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Intermediated Family Contracts
February 2016
With French Summary
PROVISIONAL VERSION
Robert J. Gary-Bobo
CREST, ENSAE
Meryam Zaiem
CREST, ENSAE
15 rue Martel• 75010 Paris • France • Tel.: +33 (0) 1 43 12 58 00 • Fax: + 33 (0) 1 43 12 58 01
E-mail: [email protected] • www.oee.fr Siret: 424 667 947 00024
Contrats familiaux intermédiés
29 février 2016
Résumé en français de l’article de Robert Gary-Bobo et Meryam Zaiem, CREST, ENSAE.
Résumé court
L’article étudie la possibilité et les propriétés de contrats familiaux intermédiés. Ces contrats sont
définis comme des arrangements de famille, aboutissant au partage des revenus et de la richesse au
sein d’un groupe familial, formalisant un plan inter-temporel, contingent à des événements aléatoires,
comme le décès ou la maladie d’un parent. Ces contrats peuvent jouer un rôle déterminant en
anticipant les successions, mais ont la particularité de faire intervenir un intermédiaire financier, à la
fois banquier et assureur. Le contrat familial intermédié est un bouquet de contrats bilatéraux, signés
entre la banque et les membres d’une famille au même moment ; c’est aussi un paquet de produits
financiers et d’assurance vendus simultanément aux clients de la banque, tendant à résoudre un
problème d’accord familial et de succession. Ce paquet de produits regroupe des crédits hypothécaires
classiques (aux enfants), des prêts viagers hypothécaires (aux parents) ou des achats en viager par la
banque elle-même, des conversions en rente viagère, de l’assurance-décès (sous une certaine forme)
et des transferts implicites entre parents et enfants. Suivant les pays et les contextes réglementaires et
légaux, le contrat familial intermédié est aussi un instrument d’optimisation fiscale. L’article étudie
les propriétés d’un contrat familial optimal sous trois hypothèses-clef : l’aversion pour le risque des
clients ; l’existence d’un certain degré d’altruisme des parents (à l’égard de leurs enfants) ; un fort
attachement des personnes âgées à leur logement. Des simulations évaluent la profitabilité de tels
contrats pour le banquier. Le contrat peut s’adresser à des personnes dont les revenus sont de niveau
moyen, qui sont propriétaires de leur logement (ou détenteurs d’un certain patrimoine), et qui
voudraient aider leurs enfants (ou petits-enfants) à s’établir. La contribution de l’article n’est pas de
mettre en évidence l’importance potentielle du crédit viager hypothécaire pour les personnes âgées,
mais de montrer qu’un intermédiaire financier, en améliorant le partage des risques au sein d’une
famille, peut dégager un profit non-négligeable. Secondairement, l’analyse des contrats familiaux
suggère des pistes pour l’optimisation fiscale.
Résumé étendu
Introduction. L’article étudie la possibilité et les propriétés de contrats familiaux intermédiés. Ces
contrats sont définis comme des arrangements de famille, aboutissant au partage des revenus et de la
richesse au sein d’un groupe familial, formalisant un plan inter-temporel, contingent à des événements
aléatoires, comme le décès ou la maladie d’un parent. Ces contrats ont un caractère notarial, puisqu’ils
peuvent jouer un rôle déterminant en anticipant les successions, mais ont la particularité de faire
intervenir un intermédiaire financier, à la fois banquier et assureur.
Le contrat familial intermédié est un bouquet de contrats bilatéraux, signés entre la banque et les
membres d’une famille au même moment ; c’est aussi un paquet de produits financiers et d’assurance
vendus simultanément aux membres de la famille cliente de la banque, tendant à résoudre un problème
d’accord familial et de succession. Ce paquet de produits regroupe des crédits hypothécaires classiques
(aux enfants), des prêts viagers hypothécaires (aux parents) ou des achats en viager par la banque ellemême, des conversions en rente viagère, de l’assurance-décès sous une certaine forme et des transferts
implicites entre parents et enfants. Suivant les pays et les contextes réglementaires et légaux, le contrat
familial intermédié est aussi un instrument d’optimisation fiscale. L’article étudie les propriétés d’un
contrat familial optimal sous trois hypothèses-clef : l’aversion pour le risque des clients ; l’existence
d’un certain degré d’altruisme des parents (à l’égard de leurs enfants) ; un fort attachement des
personnes âgées à leur logement. Des simulations évaluent le gain en bien être apporté par un contrat
familial groupé optimal. Le contrat peut s’adresser à des personnes dont les revenus sont de niveau
moyen, qui sont propriétaires de leur logement (ou détenteurs d’un certain patrimoine), et qui
voudraient aider leurs enfants (ou petits-enfants) à s’établir. La contribution de l’article n’est pas de
mettre en évidence l’importance potentielle du crédit viager hypothécaire pour les personnes âgées,
mais de souligner qu’un intermédiaire financier, en améliorant le partage des risques au sein d’une
famille, peut offrir une valeur ajoutée non-négligeable à ses clients. Secondairement, l’analyse des
contrats familiaux suggère des pistes pour l’optimisation fiscale.
Faits empiriques. L’analyse des données européennes disponibles sur le patrimoine et les revenus des
personnes de plus de 50 ans (Enquête européenne SHARE) montre un certain nombre de faits
instructifs. Les ¾ des ménages de plus de 50 ans sont propriétaires de leur logement en Europe. Une
fraction non-négligeable de personnes âgées possède un patrimoine immobilier élevé mais des revenus
faibles, en comparaison, en raison de l’appréciation des biens immobiliers en zone urbaine,
principalement. Par ailleurs, les enfants héritent ou reçoivent de leurs parents assez tard. La moitié des
personnes qui ont reçu un transfert d’au moins 5000 euros reçoit un transfert (don entre vifs ou
héritage) des parents après l’âge de 50 ans. Si une petite proportion de personnes, non-négligeable,
devient propriétaire à l’occasion de l’héritage direct d’un bien immobilier des parents, la moitié des
ménages propriétaires environ a acquis son logement avant que le chef de famille ait atteint l’âge de 40
ans. Donc, les ménages héritent (ou reçoivent de l’aide) en moyenne 10 ans trop tard du point de vue
de leur stratégie d’investissement immobilier.
La figure 1 ci-dessus monte l’histogramme de l’âge au premier transfert reçu (don ou héritage).
Source, enquête européenne SHARE.
Le patrimoine immobilier est la forme la plus répandue de détention du patrimoine, puisque le
patrimoine financier non-immobilier médian représente 2% du patrimoine médian net. La valeur dont
disposent les familles est donc pour l’essentiel immobilisée dans les résidences des parents. Le
patrimoine d’un ménage, sous ses diverses formes, tend pourtant à décroître après l’âge de 65 ou 70
ans. D’un autre côté, on observe que les transferts (héritages ou dons entre vifs) jouent un rôle
significatif dans l’accession à la propriété en Europe. Les ménages de personnes de plus de 50 ans
observés en Europe aujourd’hui, qui ont reçu de leurs parents, sont plus souvent devenus propriétaires
lorsqu’ils ont reçu plus jeunes et lorsqu’ils ont reçu beaucoup — ce que le bon sens suggère, mais que
la statistique confirme. L’accession à la propriété n’est pas bien expliquée par le revenu du
propriétaire mais la probabilité d’accession augmente significativement avec l’existence et le montant
des dons et héritages. Il est raisonnable de penser que ces personnes, devenues elles-mêmes parents,
donneront à leurs enfants des sommes d’argent et des biens qui pourront jouer un rôle déterminant
dans la construction de leur patrimoine.
Tableau 1
% des ménages ayant
reçu don ou héritage*
Autriche
56
18
Allemagne
53
30
Suède
57
42
Pays Bas
59
28
Espagne
87
17
Italie
78
19
France
72
23
Danemark
61
37
Grèce
84
25
Suisse
52
47
Belgique
79
42
Israël
83
29
Total
70
30
Source : données Européennes SHARE; *ménages de plus de 50 ans.
Tableau 2
Valeur du
logement
Cas A
135000 €
Cas B
338000 €
Cas C
540000 €
% de propriétaires*
Valeur des
autres actifs
33750 euros
d’épargne
financière
169000 €
d’épargne
financière
Epargne
135000
Actif
professionnel
d’une valeur de
675000 €
Taux d’imposition effectif
de l’héritage (enfant)
Taux d’imposition
effectif de la donation
(enfant)
Union
France**
européenne*
Union
européenne*
France**
2,8%
7%
3%
7%
5,75%
15,7%
5%
15,7%
5,12%
12,3%
4%
12,3%
*Source : European Commission (2014), Cross Country Review of Taxes on Wealth and Transfers of Wealth. Les taux
effectifs sont des moyennes sur 27 pays. **France: on étudie le cas d’un enfant unique héritier en ligne directe, l’abattement
de 100000 euros s’applique ; le dispositif Dutreil s’applique au transfert de l’entreprise dans le Cas C.
La fiscalité des transmissions par dons entre vifs ou héritages peut bien évidemment jouer un rôle clef
dans le timing des dons parentaux et donc dans la distribution des âges d’accession à la propriété. Il y
a une diversité de dispositions fiscales en Union Européenne, mais un socle commun de règles est
partagé par un groupe central de pays.
L’héritage est taxé dans 18 des 28 pays de l’Union européenne, mais la plupart des petits héritages
sont exemptés. La fiscalité des héritages est, en proportion de la recette fiscale totale, la plus élevée en
Belgique (1,37%) et en France (0,97%). Tous les pays de l’Union européenne taxent les donations sauf
l’Autriche, le Portugal, la Suède les pays Baltes et la Roumanie. Le Tableau 2 montre des taux
d’imposition effectifs moyens qui ont été calculés dans trois cas-types.
Hypothèses, modèle, contrats. L’idée de contrat familial intermédié peut être étudiée dans le cadre
d’un modèle, relativement simple, dont on montre la flexibilité et les nombreuses possibilités
d’extension. On considère le cas typique d’une mère et de sa fille unique. La mère est veuve et
propriétaire de son logement. La fille souhaite accéder à la propriété en empruntant pour acheter son
propre logement. Un contrat de crédit classique se fonderait seulement sur les revenus de la fille et
prendrait le logement de la fille comme garantie. Cette dernière héritera un jour de la maison de sa
mère si la mère ne lui fait pas un don de son vivant.
Nous faisons ici trois hypothèses importantes. Dans une population de veuves propriétaires avec une
fille unique, les individus se caractérisent d’abord par un certain degré d’aversion pour le risque (mère
et fille sont prêtes à payer une prime d’assurance pour réduire la variabilité de leur revenu et de leur
consommation de divers biens, dont le logement). Ensuite, la mère se caractérise par un certain degré
d’altruisme, c’est à dire un certain souci du bien-être de sa fille (elle est concernée, à un certain degré,
par la taille du logement de sa fille et plus généralement par son niveau de vie). Il peut y avoir des
mères plutôt égoïstes et des mères plutôt généreuses, l’hypothèse est qu’il y a un certain degré, non
nul, d’altruisme parental. Il peut aussi exister des filles qui voudraient aider leur mère. Pour cette
raison, les choix des mères s’apparentent à (et sont traités comme) des décisions économiques
collectives qui peuvent comporter un élément de redistribution (une mère riche augmentera sa
satisfaction en donnant à une fille relativement plus pauvre). Enfin, la mère est supposée très attachée
à son logement, elle ne veut pas le quitter, et elle ne veut pas le quitter pour un logement plus petit ni
même, dans une certaine mesure, pour un logement plus grand mais différent. Cette hypothèse est
assez réaliste et semble caractériser (à des degrés divers) un grand nombre de personnes âgées.
Le contrat optimal est offert par un banquier-assureur qui pratique un calcul financier (escompte) et un
calcul d’actuaire, fondé sur les probabilités de survie de la mère (et l’évaluation rationnelle d’autres
risques). Cet intermédiaire diversifie ses risques en gérant un portefeuille de contrats familiaux
suffisamment grand. Le modèle que nous étudions met l’accent sur le problème d’assurance central
qui est ici le risque de longévité de la mère. Il y a par ailleurs des risques sur le prix futur des biens
immobiliers, des risques de chute dans la dépendance dus au grand âge, et des risques de perte (ou de
fluctuation) des revenus que nous laissons de côté pour les besoins de la clarté de l’exposé. Nous
établissons que le contrat optimal, sous nos hypothèses, consiste à partager le revenu et la richesse
totale de la fille et de la mère, au cours du temps, de la manière qui suit. La mère vend sa maison en
viager à la banque; elle reste chez elle comme usufruitière ; elle touche une pension de la banque
jusqu’à sa mort. La banque fait à la fille un crédit immobilier classique qui comporte une subvention
ou un rabais équivalent à un certain don de la mère, qui peut être échelonné au cours du temps, et sur
toute la durée du crédit. La mère a pu donner une partie du bouquet de son viager à sa fille, en
franchise de droits de mutation à titre gratuit, si la partie du bouquet transférée est inférieure aux seuils
d’exemption en vigueur. Enfin, la fille est totalement immunisée contre les conséquences d’un décès
(ou de la maladie, suivie du décès) de sa mère : son revenu net de ses remboursements à la banque ne
changent pas après le décès de sa mère ; en substance, la fille a déjà touché son héritage.
La mère et la fille auraient-elles pu mettre en œuvre cet arrangement et « régler leurs affaires » par
leurs propres moyens, en combinant des outils classiques ? Pas tout à fait, ni si facilement. La mère
aurait pu vendre son logement en viager de gré à gré, avec les risques et difficultés que cela comporte.
L’entité banque-assurance, gérant un portefeuille immobilier, serait beaucoup plus efficace dans ce
domaine. Elle peut ensuite transférer tout ou partie de son bouquet ou de sa rente viagère, en évitant
les impôts sur les dons entre vifs, au besoin en donnant du liquide chaque mois à sa fille pour l’aider à
payer son crédit immobilier. La fille aurait pu prendre un crédit immobilier seule, mais pas aussi
facilement, ni dans des conditions aussi bonnes, car la richesse de sa mère ne serait pas prise en
compte. Enfin et surtout, l’aide de la mère cesserait avec sa mort, ce qui n’est pas le cas avec le contrat
familial intermédié, car le banquier qui est aussi assureur sur la vie, peut aider la fille après la mort de
la mère. C’est un avantage-clef de l’intermédiation.
Simulations. Nous calculons le contrat optimal pour un échantillon artificiel de familles, calibré pour
être représentatif d’une classe moyenne aisée de propriétaires-occupants vivant en région parisienne, à
titre d’illustration (micro-simulation). On calcule pour chaque famille le profit maximum (en termes
actuariels, espérés et actualisés) qui peut être réalisé et partagé entre les parties à l’occasion de la mise
en place du contrat familial optimal. On montre que la profitabilité de ces contrats est positive et nonnégligeable ; que la fille achète un logement en moyenne plus grand qu’en l’absence de contrat
familial ; que la satisfaction (utilité ou bien-être) de la famille augmente. Pour calculer le profit
maximum, on compare l’utilité de la mère sous contrat familial avec son utilité si elle ne peut disposer
que d’outils classiques (option externe de la famille). Cela permet de calculer le surplus actualisé
maximum qui peut être extrait d’une famille et d’étudier sa distribution dans la population.
Passage à la pratique. On voit bien que ce type d’arrangement est potentiellement plus intéressant
pour les familles s’il comporte une assurance contre le risque de dépendance, permettant, par exemple,
de lisser les frais d’une maison de retraite de bon confort, où ceux d’une assistance à domicile. De
même, il peut aussi être employé pour résoudre, avant la mort des parents, un problème de partage
entre frères, sœurs et parent survivant. Enfin, le contrat peut « sauter » une génération et servir à aider
les petits enfants. Ce type de contrat peut aussi être adapté pour aider un enfant ou petit enfant à
monter un commerce ou une entreprise, et pour financer les études d’un petit enfant. Ces derniers cas
méritent une réflexion spécifique. L’articulation de l’arrangement familial avec l’assurance-vie telle
qu’elle se pratique en France mérite également une réflexion spécifique. L’intuition que nous
soumettons à la réflexion du lecteur, c’est que les obstacles comportementaux, les réticences à la
souscription de contrats viagers (énigme de la sous-annuitisation), le problème du recours limité au
viager ou aux prêts viagers hypothécaires de type anglo-saxon, peuvent être plus aisément levés ou
atténués dans le cadre de la définition d’un contrat familial intermédié. La façon de présenter ce type
de contrat et ses avantages à la clientèle (framing) mérite une étude particulière.
Intermediated Family Contracts∗
Robert J. Gary-Bobo, Meryam Zaiem
29 February 2016
Abstract
We study intermediated family contracts. These contracts are used to share the family’s wealth and income among its members and to transmit wealth from one generation
to the next, with the help of a banker-insurer, according to a multi-period plan. Family
arrangements can be defined as a bundle of bilateral contracts signed simultaneously with
a bank by several members of a family. These contracts may be contingent on a number
of publicly observable events, like a person’s death; they can also be analyzed as a bundle of financial products, combining equity release and home loans (reverse and standard
mortgages), annuitization (longevity insurance), consumption smoothing and long-term care
insurance. A secondary benefit of such a product may be tax optimization. The properties
of optimal intermediated family arrangements are studied under three main assumptions:
parents and children are risk-averse; parents are altruist to a certain degree; the elderly are
owner-occupiers and strongly attached to their homes. We used calibrated simulations to
study the welfare and comparative statics properties of the optimal contract in a population
of families. Simulations provide bounds on the surplus that can be extracted by the intermediary, by means of a family contract, i.e., on the profitability of a given family for the
banker-insurer.
Keywords : Bequests; Aging; Annuities; Inter-vivos transfers; Reverse Mortgages; Taxation; Housing equity.
∗
CREST, ENSAE, 15 boulevard Gabriel Péri, 92245 Malakoff cedex, France; E-mail:[email protected]
1
1
Introduction
In the present article, we study intermediated family agreements. These contracts are used to
share the family’s wealth and income and to transmit wealth from one generation to the next, with
the help of a banker and (or) an insurer. The family arrangements that we study can be defined as
a bundle of contracts signed simultaneously with a bank by several members of a family. In legal
terms, it may not be necessary to sign a multilateral agreement, but the banker may require that
the family members involved sign a two-sided contract with the bank at the same moment, and
the contracts may be contingent on a number of publicly observable events, like a person’s death.
The family contracts can also be analyzed as a bundle of financial products, combining equity
release and home loans (reverse and standard mortgages); annuitization (longevity insurance),
consumption smoothing and long-term care insurance. Intermediated family contracts solve
several important problems at the same time. The most important of these problems is the
specification of an intertemporal plan to share the parents’ estate with the children. This is why
these contracts could also be called notarial. A secondary benefit of such a product may be to
save on some taxes.
We study these intermediated notarial contracts under several key assumptions. First, the
older member of the family (hereafter called the mother ), is supposed to be an altruist. The
mother wants to maximize a weighted sum of her utility and the utilities of her children. Second,
a number of risks, and in particular, the mother’s longevity risk and the mother’s risk of a severe
health accident requiring long-term care, as well as market risks like house-price variability, can
be diversified and mutualized by a banker-insurer, who holds large bundles of family contracts.
Third, we assume that the mother owns her residence and that she is strongly attached to her
house: the disutility of moving or the disutility of house downsizing are supposed to be very
high. The banker can therefore also offer a liquidity service, unlocking the mother’s housing
equity by means of a reverse mortgage, a home reversion plan or a French viager contract, while
permitting the mother to stay in her home. The money released can be used to schedule gifts to
the next generation, and the children can at the same time borrow to buy their own homes.
We characterize the set of optimal intermediated family contracts in a simple model with a
finite horizon, discrete time, risk-averse and altruist agents. The optimal solution may depend
on the tax system, but if the tax rate on bequests is positive, the mother’s home equity should
be entirely liquidated by means of a viager contract (or home reversion plan); direct inter-vivos
gifts to the children should be zero but the children’s mortgage payments can be subsidized;
the mother’s assets can simultaneously be annuitized and the children are fully hedged against
the mother’s longevity risk. The model can be generalized to study various related questions.
In particular, the analysis can be extended to study the case of multiple children or heirs and
multiple risks.
We then calibrate the model to explore the properties of family contracts by means of numerical simulations. We discuss the welfare gains of these contracts, and show how the surplus can be
shared and extracted by financial institutions. The simulations are calibrated using French data
2
on income and wealth, and we focus on seniors living in the Paris region as an illustrative example. Our simulations provide an evaluation of the profitability of improvements in the efficiency
of risk-sharing due to the intervention of financial intermediaries in family arrangements.
Finally, we discuss the practical problems posed by the design of intermediate family contracts, focusing on the potential behavioral obstacles to the marketing of such products. The
contribution of the paper is not to emphasize the usefulness of equity release instruments, that
has been studied by various authors (see below), but to explore the consequences of optimal
risk-sharing between family members in terms of product design (and the associated financial
engineering).
The present paper starts with an empirical section in which we use the European SHARE
survey on the elderly in Europe to show a number of statistical facts related to the transmission
of wealth within families.
Literature
The life-cycle theories of consumption and savings have generated a number of puzzles. In
particular, the observed behavior of the elderly seems to contradict standard theory, challenging
the individual rationality assumptions. One of the well-known puzzles is the absence of asset
decumulation, or dissaving, among retired and aging households. It seems that the elderly do not
wish to downsize their housing consumption and do not even seem to be willing to borrow against
home equity to finance consumption. These facts have been discussed a long time ago. Except
for Social Security and some pension-related assets, housing equity is the most important asset
of a large fraction of older citizens in Europe and in the United States. Venti and Wise (1989),
(1990) showed that the elderly are as likely to move to a larger house as into a smaller one. This
seems to contradict the prediction of life-cycle theory according to which old agents should reduce
savings and housing consumption. In general, retired households resist financial downsizing in a
way that seems “irrational” (see Dynan et al. (2004), De Nardi et al. (2010)). Further study of
this question shows that, in the absence of changes in household structure (for instance, the death
of a spouse), or other shocks, like the need to enter a nursing home, most elderly families are
unlikely to move (see Venti and Wise (2004)). It is reasonable to assume that retired individuals
are strongly attached to their homes. This deep-rooted attachment has been studied by social
scientists (see for instance Curry et al. (2001)). Psychological factors certainly play a rôle, as
introspection may reveal, but home-ownership is also objectively providing a form of insurance
against various risks. Firstly, ownership is a hedge against upward variations in future house
prices; secondly, liquidating the house and entering the rental market exposes the household to
rent variability, and this latter risk cannot be insured. Sinai and Souleles (2005) have shown
that home-ownership rates are higher in areas with greater variability of rental prices. In his
comments on Venti and Wise (2004), Jonathan Skinner (2004) recalls that the most common
reason for moving is poor health, but that the second most common reason is to move closer to
family (see Choi (1996)). This may explain why moving does not necessarily cause downsizing.
Skinner (1996) also discusses the role of (psychological and real) moving costs.
It seems relatively well-established that the commitment of the elderly to their homes is also
3
due to the fact that these homes are a store of value, and more precisely, a substitute for longterm care insurance. There exists a literature on the reasons for which long-term care insurance is
underdeveloped (see, e.g., Brown and Finkelstein (2007)). Medicaid and, more generally, public
insurance systems may be responsible for some crowding out of the demand, but the main reason
seems to be the fact that home-ownership plays the role of long-term care insurance for many
people (see Thomas Davidoff (2010)).
Venti and Wise (2004) put the emphasis on another important fact: the apparently limited
recourse to housing equity-release instruments, and mainly to reverse-mortgage contracts that
are available in the United States since the mid 90s. Many older households have substantial
wealth locked in illiquid housing. They should normally like to release this wealth to finance
consumption. Reverse mortgages in the US, the UK and other countries (to a certain extent,
viager contracts in France) attract a limited number of clients only, in spite of important potential
benefits — at least in theory. In many countries, there is a significant segment of “incomepoor” and “house-rich” individuals. For international comparisons of housing equity adjustment
decisions, see Chiuri and Jappelli (2010), Fornero and Rossi (2012). As mentioned by Skinner
(2004), Davidoff (2010) and others, the limited recourse to reverse mortgage loans may not
be irrational, but, on the contrary, may be due to several clearly understandable factors: the
insurance properties of home-ownership on the one hand, and tax optimization on the other
hand, since real property is a “safe haven” under some tax systems.
This does not mean that behavioral economics considerations do not contribute to the explanation of some facts: financial literacy considerations (and risk aversion) have been used to
explain the limited recourse to reverse mortgages (again see Fornero and Rossi (2012); on mortgages in general, see Cox et al. (2015)). Because of imperfect insurance against expenditure
shocks due to health accidents, the elderly may prefer a well-designed line-of-credit plan to a
housing-equity release contract (Fratantoni (1999)). The stylized fact that equity release loans
are characterized by low take-up rates may have to be revised because the reverse-mortgage
market witnessed substantial growth in the mid 2000s, particularly in the United States (see
Hui Shan (2011), Sinai and Souleles (2008)). Higher house prices, generating windfall gains for
many homeowners, may finally persuade large fractions of the potential borrowers to engage
into equity release (on the rise of house prices, see Gleaser et al. (2005)). Recent research will
lead to improvements in financial engineering techniques and, among other things, to a better
understanding of the borrowing limits of reverse mortgage contracts (see, e.g., Pu, Fan and Deng
(2014)). In some countries, such as France, the reverse mortgage market is totally underdeveloped. The French prêt viager hypothécaire has been legalized by an act passed in 2006. This
kind of loan is strictly regulated and French banks cannot advertise it. In contrast, the viager
contract is not limited.
The degree of financial development of a country may be highly correlated with the degree
of financial literacy of its inhabitants. Economic agents typically make mistakes and do not
behave as predicted by normative theory, but when explanations are offered to them, in general,
they understand the benefits of new products. Even if they don’t understand the contracts well,
4
at some point, they simply start to mimic successful neighbors who did choose these products
before them. But either kind of learning takes time. For that reason, the development of reverse
mortgages may become impressive in a few years from now, even in financially under-developed
countries, simply because they are useful and enhance welfare, as predicted by normative theory,
and since people will eventually understand their usefulness. Note that limits to the development
of equity-release contracts can also be due to the presence of bequest motives, that we discuss
below.
The French market for housing-equity release is still in its infancy. In this country though,
the viager contract is a traditional mode for the sale of a house. Yet, this market is residual
because individual buyers have to bear the longevity risk of the seller, which is a typically suboptimal way of sharing risks. These risks should obviously be borne and diversified by insurance
companies and banks. In France, financial intermediaries started to buy houses in exchange for
a viager contract in the recent months. An experimental Real Estate Investment Trust (i.e.,
Certivia) has been launched in 2014, under the aegis of the Caisse des Dépôts, with limited
amounts of capital; the private financial sector has just started to intervene as an intermediary
in the viager market, with a few newly created funds, that is, more precisely, with FPCIs (in
French, Fonds Professionnel de Capital Investissement) like Silver Estate. A few economists
advocated the development of intermediated viager markets in France (see Masson (2014)). The
social benefits of this type of financial innovation could indeed be huge, and to the best of our
knowledge, they have not been estimated by means of rigorous methods. The economic literature
on the French traditional viager market is recent and, to the best of our knowledge, papers on
this topic are rarities (see Février et al. (2012)).
The annuity puzzle is another important behavioral anomaly (see, e.g., the survey of Benartzi
et al. (2011)). Older households do not annuitize their wealth, while theory predicts that they
should. The classic result is due to Yaari (1965): a risk-averse agent without a bequest motive
should completely annuitize her wealth, that is, sell her assets in exchange for a sequence of
certain payments that will be terminated after the end of the agent’s life. The use of annuities
is rational because they protect old agents against the risk of outliving their income. In the
economics of life insurance, those who live longer than the average receive a subsidy from those
who die early. Davidoff et al. (2005) present conditions for the optimality of full annuitization
under market completeness which are more general than Yaari’s assumptions. In incomplete
markets, they show that consumers will generally want to annuitize a significant fraction of their
wealth. There is also a debate on the reasons for which these markets are underdeveloped, and
this is still an open research question (again, see Benartzi et al. (2011)). Recent research has
focused on behavioral obstacles to annuitization. The behavioral theories of Kahneman and
Tversky (Khaneman (2011), Kahneman and Tversky (1979)) and Richard Thaler (see Thaler
(1999)) have been applied to understand the annuity markets (see, e.g., Hu and Scott (2007)).
Scott, Watson and Hu (2011) study annuity design problems and suggest that some innovations
may increase participation in the annuity market and allow for significant welfare improvements.
Again, bequest motives and the desire to keep real estate assets as a hedge against large shocks,
5
as well as the rigid or irreversible consequences imposed by a limited set of available options may
explain incomplete annuitization and limited participation.
On these themes, see Campbell (2006), who argues that if some observed facts are unquestionably behavioral anomalies in a certain sense, their origin is often unclear. It may be that
theory, and the economic models, are too simple to capture the complexity of individual decision
problems; the data sources are too limited to convincingly identify some effects; and of course,
some —but not necessarily a majority of— individuals lack knowledge and simply make mistakes.
It follows that the data may lead to rejection of a textbook model but that a more sophisticated
theory relying on rationality assumptions would fit the data reasonably well. Some key parameters in life-cycle and financial theories are typically hard to identify in the econometric sense.
The environment of households is constantly changing, in particular because of innovations and
due to the development of financial markets, and individuals may be learning slowly, or many
of them just imitate the behavior of others, so that adjustment to economic equilibrium in the
standard sense takes time. We keep these difficulties in mind in what follows.
The next important point in the following analysis is the existence of parental altruism and
bequest motives. Economists hesitate about the observed bequests: are they random, purely
unintentional transfers or do they reflect the existence of a bequest motive? It is likely that
observed bequests are a mixture of intentional and unintentional transfers of wealth. Without
bequest motives, the life-cycle theories predict that bequest will be nonzero, because death is not
perfectly predictable. A further question is the importance of individual mistakes, behavioral
biases and inaccurate planning in the distribution of inheritance. Economic theory then hesitates
about the appropriate model for the bequest motive. In a nutshell, is it better to assume that
individuals’ preferences are characterized by “joy-of-giving”, that is, an ad hoc utility for the gifts
made to children, or should we model individuals as pure altruists? In this context, pure altruism
means that utility functions depend on the utility of children, leading to dynastic and recursive
formulations of utility, since the utility of the father depends on the utility of the son which in
turn is a function of the grandson’s utility, etc.
The importance of the “joy of giving”, altruism and bequest motives has been recognized a
long time ago by economists (see, e.g., Becker (1974), Andreoni (1989), Abel and Warshawsky
(1988), Laferrère and Wolff (2006) ). The distinction between the various sources of inheritance
flows is of course crucial for the theory of taxation (see, e.g., among recent papers, Cremer and
Pestieau (2006), Diamond (2006)), and this may explain the failure of Ricardian equivalence. If
bequests are mainly accidents, they can be taxed with no incidence on savings.
The evidence concerning the existence of a bequest motive supports the hypothesis (see,
e.g., Kopczuk and Lupton (2007)). There also exists an important econometric literature on
family transfers and the pure altruism hypothesis. The latter assumption is rejected by some
tests (Altonji et al. (1997)), but currently, there is a lack of consensus on this question (see
Kaplow (2001), Kopczuk (2013)). Again, a possible explanation for the lack of asset run-down
and under-annuitization may be aversion to the prospect of having insufficient wealth to pay for
private long-term care, and therefore, of needing public care. Ameriks et al. (2011) formulated
6
the hypothesis that there exists a public-care aversion among the elderly in the United States.
These authors endeavored to disentangle the effects of public care aversion and bequest motives
on the degree of wealth decumulation during retirement. They found that public-care aversion is
a significant driver of precautionary savings. These fears may have equivalents in other countries.
According to Ameriks et al. (2011), there is a “powerful interaction between interest in annuities
and the institutions that provide for long-term care. (...) The demand for annuities would be far
higher if they included some acceptable form of long-term care insurance”. They also find that
bequest motives are “more prevalent and spread deeper into the middle class than is generally
believed”, and they find evidence for heterogeneity in these motives.
In the following, Section 2 provides some descriptive statistics and econometric tests, based on
European data, namely, the SHARE survey. Section 3 presents the institutional setup: the existing housing-equity release instruments and inheritance and property taxation rules (in Europe).
Section 4 presents a simplified version of our family contract model. A number of extensions
and generalizations are discussed in Section 5. Section 6 presents various numerical simulations
of the model, based on its multi-period extension. As a conclusion, Section 7 discusses contract
design and the role of behavioral obstacles.
2
Descriptive Statistics and Econometric Tests
We begin with a presentation and discussion of some descriptive statistics and econometrics based
on SHARE data. SHARE is a large scale survey of the health and wealth of persons aged 50 and
more in Europe. In the following, we particularly use the first wave of SHARE, with more than
30000 individuals surveyed between 2004 and 2005. We will focus on a number of features of the
survey that are directly relevant to our later analysis: income (pensions), wealth, real property,
intergenerational transfers (bequests and inter vivos gifts), accumulation and decumulation of
wealth.
We work with a sample of 20505 individuals, called principal respondents. Each principal
respondent is attached to a household. This is due to the double structure of the SHARE
survey: the data provide information on individuals, but also on the household to which they
belong (generally on the spouse). Some variables, (e.g., age, gender, health status) refer to the
individual called principal respondent. But other variables, like, wealth, financial assets, real
estate assets, debt, income are computed for the household attached to the principal respondent.
Among these individuals, we more particularly studied those who received money from their
parents. On the SHARE survey and methodology, see Börsch-Supan et al.(2005a), Börsch-Supan
and Jürges (2005b).
Table 1 breaks the 20,505 households by country of residence. In this sample, among principal
respondents, there are 10,732 women (slightly more than 50% of the sample). Table 2 gives the
marital status of the surveyed individuals. The number of widows is of course particularly high,
since the principal respondent’s age is distributed between 50 and 103. The size of the household
is distributed as indicated in Table 3. Table 4 gives the distribution of the health status in 5
7
Table 1: Country of Residence
Count
%
Austria
1,177
5.7
Germany
1,942
9.5
Sweden
2,111
10.3
Netherlands
1,908
9.3
Spain
1,646
8.0
Italy
1,732
8.4
France
2,033
9.9
Denmark
1,163
5.7
Greece
1,972
9.6
685
3.3
Belgium
2,498
12.2
Israel
1,638
8.0
Switzerland
Source: SHARE data set, first wave, principal respondents only.
categories (from very good to very bad).
We now study the housing variables. Each principal respondent has a principal residence.
Table 5 gives the distribution of the housing status. In the sample, 70% of the households are
owners of their principal residence. Table 6 explains how the residence was acquired. We see
that the house was received as a bequest or a gift in 10% of the cases.
To obtain a better understanding of access to the status of owner-occupier, a glance at the
distribution of the age at which the residence was acquired is useful. Fig. 1 gives the histogram
of this distribution, based on SHARE data. Strictly speaking, Fig. 1 gives the probability of
acquisition at an age denoted th , given that the age a at the moment of the survey is greater
than 50 and given th ≤ a. It follows that the reported frequencies are biased estimates of the
probabilities of acquiring a home at ages th > 50, because data are censored for individuals of
age a > 50 that would buy their homes only in the future, at an age th > a. The associated
survival distribution of the acquisition age could be estimated by the Kaplan-Meier method,
but the estimates of probabilities would be imprecise for ages greater than 50. The reader
must remember that probabilities of acquisition after 50 are somewhat under-estimated by the
histogram frequencies. We see that roughly 60% of the studied sample bought their house
between 30 and 50, while 20% became owners before 30.
We now look at transfers and try to see if the data suggests that transfers played a role in
helping the households building up their wealth. The data doesn’t indicate if the money received,
or the assets received by households take the form of a bequest or of an inter-vivos gift. This is
why we describe gifts and bequests simply as transfers below. These transfers are concentrated.
Most of the surveyed individuals never received a transfer greater than 5000 euros.
8
Table 2: Marital Status
Count
%
12,501
61.0
registered partnership
302
1.5
married, living separated from spouse
298
1.5
never married
1,409
6.9
divorced
1,749
8.5
widowed
4,233
20.7
married and living together with spouse
Table 3: Size of the Household
Count
%
1
6,018
29.3
2
9,785
47.7
3 or more
4,702
22.9
Table 4: Health Status
Count
%
very good
2,924
14.3
good
6,071
29.6
fair
6,944
33.9
bad
3,404
16.6
very bad
1,146
5.6
Table 5: Ownership and Tenancy Status
owner
member of a cooperative
tenant
subtenant
rent free
usufructuary (France only)
Count
%
14,307
69.8
463
2.3
4,563
22.3
74
0.4
1,071
5.2
14
0.1
9
Table 6: How the residence was acquired
Count
%
purchased or built solely with own means
11,612
78.8
purchased or built with help from family
1,009
6.8
received as a bequest
1,446
9.8
received as a gift
201
1.4
acquired through other means
463
3.1
0
.01
.02
Density
.03
.04
.05
Figure 1: Histogram of age at which house was acquired
0
20
40
60
Age house was acquired
10
80
100
Table 7: Share of households who received a transfer
% owners % households who received a gift or bequest
Austria
56
18
Germany
53
30
Sweden
57
42
Netherlands
59
28
Spain
87
17
Italy
78
19
France
72
23
Denmark
61
37
Greece
84
25
Switzerland
52
47
Belgium
79
42
Israel
83
29
Total
70
30
Source: SHARE data set, first wave.
We found 6274 individuals who received a transfer greater than 5000 euros at least once in
their life. There are various missing data problems: for instance, we sometimes do not know
when the transfer was received. To study the timing of the transfer (when it was received),
and the amount, we selected the largest sub-sample without missing observations of the relevant
variables.
Table 7 shows the proportion of owners and the percentage of those who received at least
one transfer, broken down by country. The European average shows that 70% of the SHARE
households are owner-occupiers and that 30% received a transfer. Fig. 2 breaks the population
by country of origin and shows the boxplots of the distribution of gifts, countries being ranked
by the median transfer, in ascending order. There are many outliers the values of which are
not depicted. The figure shows striking differences between some countries — for instance,
between Sweden and Switzerland. Fig. 2 also shows that dispersion (inter-quartile range) varies
widely from one country to the next. It seems that richer countries and(or) countries with less
redistribution by taxation have larger medians and a much larger dispersion of their distribution
of transfers — with inheritance-tax havens at the bottom of the list.
The survey records a list of at most five transfers per respondent with the time at which they
were received. We now look at the distribution of the age at which the principal respondent
received the first and the second transfers (if they exist). Transfers are ranked by magnitude,
but the data can be rearranged to provide a time sequence. Figure 3 depicts the distribution
of the age at which the first transfer (not necessarily the greatest) was received. In this figure
and the next, the same caveat is needed: the displayed frequencies are biased estimates of the
corresponding probabilities, because data is censored, for persons with ages strictly greater than
11
Figure 2: Boxplot of total transfers, by country
Spain
Sweden
Denmark
Netherlands
Italy
Greece
Israel
Belgium
France
Germany
Austria
Switzerland
0
100000
200000
300000
Total amount of transfers
400000
500000
excludes outside values
50 at the time of the survey may receive a transfer in the years following the survey (but KaplanMeier survival estimates would not be very precise for old ages).
On Fig. 3 we see that people receive help from their parents at 50 on average. The median
age is around 50 (probably slightly overestimated). In addition, 75% of the individuals receive
their first transfer after 42 (at least). We can safely conclude that help coming from the parents
or bequests arrive late in life. Presumably, young adults would have needed help during their
thirties. Reality shows that they have to wait until they themselves have raised their children
and bought a house before receiving a transfer that is very likely to be a bequest. It is not
easy to know which of the transfers reported in the data are in fact bequests, when the data is
missing. When the individual’s parents are still alive, we know that the transfer was in fact a
gift. Figure 4 gives the observed frequency distribution of the age at which the second transfer
was received (second in historical order). The median age of the second transfer is slightly larger,
at 54. Its standard deviation is smaller than that of the distribution reported on Figure 3 (10
years instead of 12.5), and 75% of the individuals receive their second transfer after 48 (at least).
Again, the distribution gives a slightly biased view of the corresponding probabilities because of
censoring. In the sample, we also observe that 81% of the individuals receive only one transfer
above 5000 euros and 97.5% receive only two such transfers. This is why we focused on the first
two transfers. Presumably, the vast majority of all recorded transfers are bequests.
We now study the age at which the main property was acquired as possibly determined or
caused by the transfers. Figure 1 depicts a frequency distribution where 99% of the mass is above
19 years old, the mean is 41 and the median is 39. The typical individual buys a home (at least)
10 years before receiving a bequest from his(her) parents (with the caveat due to censoring).
12
0
.01
Density
.02
.03
.04
Figure 3: Age at which first transfer was received
-50
0
Age at 1st transfer
50
100
0
.01
.02
Density
.03
.04
.05
Figure 4: Age at which second transfer was received
20
40
60
Age at 2nd transfer
13
80
100
0
.02
Density
.04
.06
Figure 5: Distribution of distance between house acquisition and first transfer
-100
-50
0
Distance in years between acquision and 1st gift
50
Source: SHARE data set, first wave. Distribution of
distance, in years, between the year of house acquisition and the year the first transfer was received
0
.01
Density
.02
.03
.04
Figure 6: Distribution of distance between house acquisition and second transfer
-40
-20
0
20
40
Distance in years between acquision and 2nd gift
Source: SHARE data set, first wave. Distribution of
distance, in years, between the year of house acquisition and the year the second transfer was received
14
60
Table 8: Summary statistics about wealth
Mean
Median
Standard deviation
Total value transfers
87,100
37,089
149,815
Financial assets
55,189
12,000
108,350
Total assets
317,932
200,000
451,700
Debt
26,476
0
56,423
Net wealth
282,679
174,100
435,549
Source: SHARE data set, first wave. Figures in 2004
euros.
We now consider the possible relationship between inter-generational transfers and property
acquisition. Figure 5 depicts a remarkable distribution, namely that of the difference (in years)
between the main residence’s acquisition date and the moment of the first transfer. The distribution displayed on Fig. 5 is obviously the mixture of a mass at point 0 (hereafter called the
“peak”) and a regular bell-shaped density. The distribution’s mean is 9 years, with a standard
deviation of 15 years, and 50% of the individuals wait at least 7 years, after the transfer, before the acquisition of their property. Figure 6 shows the equivalent distribution for the second
transfer, and there is no peak. The latter distribution is less “regular”.
It is now interesting to study what characterizes individuals in the peak of Fig. 5. Out-of-peak
and within-peak individuals are strikingly different. In essence, 83% of the off-peak individuals
purchased or built their houses “solely with their own means”, while 50% of the within-peak
individuals in fact obtained their property directly as a bequest. Only 7% of the individuals
declare that they purchased their home with the help of their family.
We will come back below to the study of transfers but we first describe the distribution of
income and wealth in the sample. Table 8 gives the medians and means of several important
variables: the total value of transfers; the value of a household’s financial assets; the household’s
total assets (including real estate); the household’s debt and the household’s net worth. It is
immediate that debt is almost negligible, as a first approximation, for the vast majority of the
households. The median value of debt is zero. Median financial assets are also small in proportion
to total assets: housing is the only important asset for most people. The median transfer is not
negligible and it may cause (or simply be correlated) with the acquisition of assets (real estate).
The dispersion of financial assets and total assets is huge, reflecting a number of well-known facts
about inequality.
We will learn more about inequality among seniors in Europe by looking at boxplots of the
key variables related to wealth, by country. These statistics are given by Fig. 7, in which
the medians have been ranked in ascending order. Table 9 gives the corresponding figures and
permits one to compare the medians and the means of the key financial variables. Fig. 8 gives
another view of the key variables, averaged over all survey countries but broken by size of the
household.
15
Figure 7: Total assets, debt and net worth, by country
Austria
Germany
Greece
Sweden
Spain
Israel
Denmark
Italy
France
Belgium
Netherlands
Switzerland
0
500000
1.0e+06
1.5e+06
Total assets
Austria
Germany
Netherlands
Spain
Italy
France
Greece
Switzerland
Belgium
Israel
Denmark
Sweden
0
50,000
100000
150000
Debt
200000
250000
Sweden
Denmark
Germany
Austria
Greece
Israel
Spain
Netherlands
Italy
France
Switzerland
Belgium
-500000
0
500000
1.0e+06
Net wealth
Source: SHARE data set, first wave. Figures in 2004 euros.
16
Table 9: Means and medians of total assets, debt and net worth
Total assets
Debt
Net worth
Mean
Median
Mean
Median
Mean
Median
Austria
144,930
75,900
3,996
0
139,932
75,000
Germany
181,040
83,000
10,393
0
165,403
70,000
Sweden
181,002
90,160
32,047
4,357
142,407
59,911
Netherlands
253,286
166,800
33,178
0
214,346
106,175
Spain
214,820
104,000
4,606
0
207,027
101,000
Italy
226,387
121,000
2,963
0
220,529
120,000
France
254,343
150,000
9,593
0
239,917
125,000
Denmark
239,515
120,987
43,437
3,226
177,010
65,871
Greece
146,872
90,000
2,726
0
143,464
88,000
Switzerland
424,383
260,722
57,800
0
335,958
137,857
Belgium
220,073
150,000
4,091
0
210,175
147,600
Israel
179,120
104,895
7,495
0
170,322
98,252
Total
212,447
110,288
14,806
0
190,630
96,790
The statistics for each variable are computed with the help of
the maximal sample, giving missing observations. It follows
that sample size varies from one column to the next. These
computations do not rely on SHARE survey imputations (on
this question, see Börsch-Supan and Jürges (2005b)).
17
We now turn to the overall distribution of real property values, total assets, debt and net
worth. Net worth, or net wealth, is defined as the sum of all assets, corporate shares and real
estate minus debts. The vast majority of wealth values are nonnegative. The histograms of these
empirical distributions are given by Fig. 9, Fig. 10, Fig. 11 and Fig. 12, respectively. On each of
these figures, the left-hand-side plot represents the distribution of the variable (which is always
very skewed and highly dispersed, due to wealth inequality), while the right-hand side presents
the distribution of the logarithm of the same variable, in the subsample where observations of the
variable are not zero. The right-hand side plots show that the distribution of the logarithm of
nonzero observations can reasonably be approximated by a bell-shaped density. The distribution
of financial assets is dwarfed by the distribution of real estate assets. Debt too plays a minor role.
Another view of the same empirical fact is provided by Figure 13. On this figure, the horizontal
axis represents the percentiles of the distribution of a variable (i.e., total assets, net worth and
financial assets). The vertical axis reports the natural logarithm of the same variable. Fig. 13
clearly shows that debt is negligible, since the net worth and total asset inverse cdfs are very
close to each other, while financial assets represent a minor share of total assets.
Next, we study the distribution of income.1 Income is defined as the sum of labor income,
pensions, asset income and income from real estate. The means are driven by extreme observations: it follows that the medians are more informative. Figure 14 gives the boxplots of the
household’s total income by country, ranked with respect to the medians, in ascending order. To
interpret the figure, the reader must recall that this is the distribution of income among seniors
aged 50 and more. As a consequence, Fig. 14 shows that the countries in which the seniors have
more generous incomes and pensions: Scandinavian countries, the Netherlands and Switzerland,
and to a lesser extent, France. In contrast, the median senior households are relatively poorer in
Belgium, Germany, and the south of Europe. Fig. 15 plots the mean total income of a household
as a function of age (i.e., the principal respondent’s age). We see that total income strongly
decreases with age. Fig. 16 presents the distribution of income (upper plot) and the distribution
of the log-income for nonzero observations (lower plot). This figure shows a bimodal distribution
of the log-income, which looks like a mixture of bell-shaped distribution. A glance at Fig. 14
show that there are two groups of countries, the senior friendly countries (Switzerland, Netherlands, Sweden, Denmark, France) and the rest (Spain, Italy, Israel, Austria, Germany, Greece,
Belgium). In each of these two subsets taken separately, the distribution of log-income would not
be markedly bimodal. Finally, Fig. 17 gives the inverse cdf of log-income, ranging from very low
incomes (5 = ln(150)) to 60% of the sample below (10 = ln(22, 000)) in the middle and topped
with 12.5 = ln(268, 000), with even higher incomes in the last 5 percentiles. We are now able
to assess the importance of bequests and gifts received relative to household income and wealth.
Table 10 compares total income and total transfers in different European countries. Figure 18
1
There are some problems with missing data and many mistaken answers, particularly in the case of income.
We first observed that missing data (on income and wealth) is significantly correlated with age but that this
correlation is very weak. To be more precise, an additional year induces a 0.0005 increase in the probability of
non-response for income. The corresponding figure is 0.001 per additional year of life in the case of wealth. We
find that non-response on income and wealth is also correlated with gender. Females have a higher non-response
rate; the probability of missing observation is 0.01 points higher than that of males. There is no detectable
correlation of non-response with the country of origin.
18
Figure 8: Wealth by size of the household
1
2
3 or more
0
200000
400000
Total assets
600000
800000
1
2
3 or more
0
10,000
20,000
30,000
Debt
1
2
3 or more
-200000
0
200000
Net wealth
400000
600000
19
0
0
.1
2.0e-06
.2
Density
4.0e-06
Density
.3
.4
6.0e-06
.5
8.0e-06
Figure 9: Distribution of real estate
0
500000
1000000
1500000
Total real estate
2000000
2500000
0
10
Log total real estate
20
30
10
Log total assets
20
30
Density
.2
0
0
.1
2.0e-06
Density
4.0e-06
.3
.4
6.0e-06
Figure 10: Distribution of total assets
0
1000000
Total assets
2000000
3000000
0
Density
0
0
.1
5.0e-05
Density
.2
1.0e-04
.3
1.5e-04
Figure 11: Distribution of debt
0
50000
100000
Debt
150000
200000
250000
-5
20
0
5
Debt
10
15
0
0
.1
5.0e-07
Density
.2
Density
1.0e-06 1.5e-06
.3
2.0e-06
.4
2.5e-06
Figure 12: Distribution of net worth
-4000000
-2000000
0
Net wealth
2000000
4000000
0
10
Log net wealth
20
30
-5
0
5
10
15
Figure 13: Inverse cumulative distribution functions: total assets, net worth and financial assets
0
.2
.4
.6
Log net wealth
Log financial assets
.8
Log total assets
SHARE data set, first wave. The horizontal axis represents
the percentiles of the distribution of a variable (i.e., total
assets, net worth and financial assets). The vertical axis
reports the natural logarithm of the variable.
21
1
shows the ratio transfer/income in two groups of countries (northern vs. nouthern countries).
The southern countries are Italy, Spain, Greece, Israel and France. These statistics have been
computed for the subsample of households who received a positive transfer (and a positive income). The median for this ratio is 1.60 in northern countries and 1.62 in southern countries (the
means are much higher, due to skewness and inequality and typically non-robust). Transfers are
therefore substantial in the sense that the median ratio indicates a bequest representing 1.6 years
of income. The same exercise can be applied to the ratio transfers/net worth. This is shown
by Fig. 19. The median ratio is 0.28 in the northern countries and 0.22 in southern countries.
Again, this is non-negligible. We consider the sub-population of individuals who reported both
a positive amount of wealth and a positive income (10747 individuals), and we study the time
profile of wealth as a function of age and income, to test for the presence of asset decumulation.
The OLS regressions presented in Table 12 are of the following form,
log(Wealth) = αAge + β(Age)2 + γ log(Income) + Controls + Random error.
It is striking that we find significant and precisely estimated values of α and β and γ, all significantly different from zero at the 1% level. The maximum of wealth is reached when age is equal
to −α/2β. In Table 12, column (1), we find the simplest regression of total wealth on age, age
squared and income, and we find the striking (and reassuring) result that −α/2β = 64. This
finding is relatively robust. In column (2) of Table 12, we introduced dummies for the so-called
southern countries (France, Italy, Spain and Greece) and interactions of the coefficients on age
and age squared with the southern-country indicator. The only significant thing that we find
is that net wealth is smaller in these countries, but the wealth decumulation age remains above
to 62. The computations of the decumulation threshold age are given by Table 11. If we break
down wealth in financial assets and real estate, results are similar. The simplest regression of
financial wealth on age and income yields a decumulation age −α/2β = 71. Results are more or
less the same if we add controls, in Table 12, column (4).
Regressions of real-estate on income, age and age-squared yield a maximum at age 64 (Table
12, column (5)) or 67 with controls for southern countries (Table 12, column (6)). See also
Table 11. Further robustness tests should be performed, but it seems that some form of asset
decumulation is taking place among persons aged 65 and more. If this is indeed the case,
decumulation is non-negligible since the estimated values of β imply a decrease in wealth around
13% during the 10 years between ages 65 and 75.
It may be that some cohort effects lead to a certain degree of overestimation (or underestimation) of the decumulation effect. But this bias is likely to be limited. This point deserves further
investigation, but it is easy to see why the bias is likely to be small. Assume for simplicity that
the above regression model applies to a panel of individuals. In a regime of balanced growth,
income and assets would all grow at the same rate, say ρ. Let x denote the age of an individual.
The assets and incomes of older individuals should be inflated to correct for the possible cohort
effects. Assume that the assets of a person of age x are chosen to be A(x) = Aeρx (instead of A)
22
Figure 14: Total income by country
Spain
Italy
Israel
Austria
Greece
Germany
Belgium
France
Switzerland
Netherlands
Sweden
Denmark
0
100000
200000
300000
Total income
0
20000
Mean total income
40000
60000
80000
Figure 15: Mean total income by age
50
60
Age
23
70
80
0
2.0e-05
Density
4.0e-05
6.0e-05
Figure 16: Distribution of income
100000
200000
Total income
300000
0
.05
.1
Density
.15
.2
.25
0
-5
0
5
Log total income
10
15
-5
0
Log total income
5
10
15
Figure 17: Inverse cumulative distribution of income
0
.2
.4
Percentile
24
.6
.8
1
Table 10: Income and transfers by country
Total income
Received transfers
Mean
Median
Mean
Median
Austria
16,930
3,623
143,436
69,466
Germany
39,135
5,600
115,142
59,208
Sweden
51,402
39,186
48,773
25,257
Netherlands
53,855
35,060
66,325
29,811
Spain
17,249
1,840
58,326
23,584
Italy
16,501
2,600
84,365
39,734
France
43,468
22,926
103,283
47,956
Denmark
73,724
49,027
54,714
26,109
Greece
19,096
4,200
84,578
41,321
Switzerland
66,898
31,465
186,189
90,675
Belgium
31,584
5,735
94,183
47,034
Israel
19,815
2,603
78,867
42,993
Total
36,210
8,884
86,318
37,884
Figure 18: Ratio transfers over income by country: boxplots
Northern countries
Southern countries
0
5
10
15
Ratio: Value of transfers/Total Income
25
20
Figure 19: Ratio transfers over net worth by country: boxplots
Southern countries
Northern countries
0
.5
1
1.5
Ratio: Value of transfers/Net wealth
2
and the income is y(x) = yeρx instead of y. Then, the above regression model becomes,
log(A) = (α0 + (γ − 1)ρ)x + βx2 + γ log(y) + controls + random error.
Table 12, column (1), yields α = 0.11, γ = 0.2. Take for instance ρ = 0.03 (a reasonable value
for a real growth rate). Then, we find
α0 = α + (1 − γ)ρ = 0.11 + 0.024 = 0.134
and the coefficient β does not change. The decumulation age would then be
−α0 /2β = 0.134/0.0018 = 74.
This back-of-an-envelope calculation shows that the real α may in fact be slightly underestimated.
If α is only slightly biased, as suggested, we found that decumulation is non-negligible, since
the estimated values of β imply a decrease in wealth around 13% during the 10 years between
ages 65 and 75. It may be that the speed of rational dissaving, should be higher when survival
probabilities are taken into account, but risk aversion and imperfect insurance may explain the
observed facts, to a large extent. Given that the regressions presented in Table 12 are nonlinear
in age, we computed a number of marginal effects drawn from regressions with effects varying
by country and with interactions. These marginal effects are displayed in Table 13.
A further robustness test is provided by regressions of log-wealth on age, age squared, logincome, marital status and health status indicators. These regressions are displayed in Table 14.
Health dummies are clearly very significant in the 3 columns of Table 14, with the expected signs
and ranking of the magnitudes. Given that we control for age, a worse health status is correlated
with smaller assets. Table 16 shows that decumulation starting around 65 is a robust finding.
26
Table 11: Age at which estate starts decreasing
Net worth
Fin. Assets
Real Estate
All sample
63
71
64
Northern countries
62
71
64
Southern countries
67
70
67
We conclude this preliminary empirical analysis of our topic with a study of the role of
transfers in the building up of assets and real property. This is done by means of simple OLS
regressions again. We estimated regressions of log-wealth on age, age squared, log-income and
the logarithm of transfers, to estimate the impact of transfers on wealth accumulation. These
regressions are displayed on Table 15. Interestingly, we find a significant, nonnegligible elasticity
of assets with respect to transfers around .3 in the three regressions (i.e., columns) of this table.
Table 16 gives the estimated age at which three measures of wealth (net worth, financial assets
and real property) start to decrease, on average.
Finally, we estimated the role of transfers in access to real property (mainly, the owneroccupier status). This is done by means of a linear probability model, regressing the ownership
indicator on log-transfers and age, age squared and controls. Table 15 displays the result of
three variants of this regression. Take for instance column (2) in Table 15. In this variant we
control for age, age squared and age at first transfer. The profile of the probability of access to
ownership is clearly a concave (quadratic) function of age, with a maximal probability at age
40. Given this, the probability of becoming an owner-occupier decreases further with the age of
the principal respondent at the moment of the first transfer. Since the log-transfer itself has a
significant and positive coefficient, we conclude that large and early transfers play a significant
role in access to ownership, more or less as expected.
We now explore the possibility of improving efficiency in the transmission of income and
wealth between parents and heirs/children, taking into account the risks associated with the
longevity of parents. This is made possible, in particular by housing equity release instruments.
27
Table 12: Net wealth and assets explained by income, age and heterogeneity by country
(1)
(2)
(3)
(4)
(5)
(6)
Dependent variable:
Net wealth
Net wealth
Fin. assets
Fin. assets
Real estate
Real estate
Age
0.1155***
0.1184***
0.1292***
0.0906***
0.0388**
0.0770***
(0.0205)
(0.0259)
(0.0230)
(0.0284)
(0.0165)
(0.0217)
-0.0009***
-0.0010***
-0.0009***
-0.0006***
-0.0003**
-0.0006***
(0.0001)
(0.0002)
(0.0002)
(0.0002)
(0.0001)
(0.0002)
0.2257***
0.2681***
0.4327***
0.4151***
0.1234***
(0.0107)
(0.0143)
(0.0121)
(0.0157)
(0.0083)
Age × Age
Log total income
Southern countries
Southern countries × Age
28
Southern countries × Age × Age
Southern countries × Log total income
0.8459
-4.0363**
2.9262**
(1.4762)
(1.6717)
(1.1594)
-0.0071
0.1039**
-0.0897***
(0.0420)
(0.0477)
(0.0334)
0.0001
-0.0007**
0.0007***
(0.0003)
(0.0003)
(0.0002)
-0.0686***
0.0078
0.1151***
(0.0216)
(0.0247)
(0.0122)
Northern countries × Log total income
0.1280***
(0.0115)
Constant
R2
Number of Observations
5.6743***
5.1505***
0.7644
2.4200**
9.4976***
8.2639***
(0.7206)
(0.9147)
(0.8060)
(0.9990)
(0.5719)
(0.7501)
0.06
0.07
0.14
0.16
0.03
0.03
10,747
10,747
8,414
8,414
9,991
9,991
Significance: *** p-value<0.001, ** p-value<0.05, * p-value<0.1.
Method: OLS
Table 13: Marginal effects for the regressions with heterogeneous effect by country
Age
Southern countries
Log total income
Number of observations
Net wealth
Fin. assets
Real estate
-0.0048**
0.0081***
-0.0004
(0.0020)
(0.0022)
(0.0017)
0.2979***
-0.4323***
-0.0247
(0.0355)
(0.0410)
(0.0275)
0.2413***
0.4179***
0.1225***
(0.0108)
(0.0122)
(0.0084)
10,747
8,414
9,991
3
Existing Institutions and Comparative Study
The present section is devoted to the description of a number of institutions that are relevant
to understand the real-world context of our formal analysis. In the following, subsection 3.1 is
devoted to equity release instruments. The latter are important building blocks of our analysis.
Subsection 3.2 is devoted to taxation of inter-vivos gifts and inheritance in Europe.
3.1
Housing equity release contracts: home reversion, lifetime mortgages and
viager
As noted in the introduction, it is well-known that many aging households keep the whole or
the most important part of their wealth locked in their homes. Real estate prices going up
in many urban areas have made many owner-occupiers more wealthy than they expected to
be. Various forms of equity release contracts could be used, in many countries, to increase the
income and pensions of the seniors. It is also well-known that the recourse to these contracts is
still limited, for various reasons. Insufficiently developed financial markets, the tax system and
behavioral problems are among the main reasons for the slow growth of these financial markets.
Another important reason for this is just the fact that these markets have been opened recently.
In the United States, the reverse mortgage programs started in 1989: this is relatively recent.
It may simply be that households have not yet fully learned about these products and their
use, because learning about these matters takes a generation or two. In continental Europe,
the underdevelopment of home equity release markets may also be due to regulation, which is
sometimes very strict. In general, these products have a bad reputation: they are viewed as
“dangerous” by regulators.
The UK, Canada and Australia have a relatively more developed equity release activity. In
the UK for instance, there exits several important types of equity release contracts: the lifetime
mortgage and the home reversion. The lifetime mortgage is a loan secured on the borrower’s
home. The loan is repaid by selling the property after the borrower’s death. The borrower retains
29
Table 14: Net wealth and assets explained by income, marital status and health
Age
Age × Age
Log total income
Health indicators
good
fair
bad
very bad
Widowed=1
Constant
R2
(1)
(2)
(3)
Net wealth
Fin. assets
Real estate
0.0930***
0.1110***
0.0322*
(0.0204)
(0.0228)
(0.0166)
-0.0007***
-0.0007***
-0.0002*
(0.0001)
(0.0002)
(0.0001)
0.1877***
0.3879***
0.1119***
(0.0109)
(0.0123)
(0.0085)
(Reference:
very good)
-0.1131**
-0.1774***
-0.0415
(0.0549)
(0.0618)
(0.0412)
-0.3607***
-0.4391***
-0.1120***
(0.0546)
(0.0616)
(0.0416)
-0.4725***
-0.7664***
-0.1079**
(0.0642)
(0.0726)
(0.0502)
-0.4842***
-0.9066***
-0.2565***
(0.0909)
(0.1056)
(0.0726)
-0.5711***
-0.4653***
-0.1790***
(0.0503)
(0.0564)
(0.0418)
6.8220***
1.9144**
9.8416***
(0.7194)
(0.8007)
(0.5754)
0.08
0.17
0.03
10,739
8,409
9,985
Method: OLS.
30
Table 15: Net wealth and assets explained by gifts
(1)
(2)
(3)
Net wealth
Fin. assets
Real estate
0.0783**
0.1029**
0.0541*
(0.0316)
(0.0411)
(0.0316)
-0.0006***
-0.0007**
-0.0004*
(0.0002)
(0.0003)
(0.0002)
0.2068***
0.3886***
0.1169***
(0.0154)
(0.0206)
(0.0145)
0.3406***
0.2755***
0.3285***
(0.0199)
(0.0261)
(0.0183)
3.7923***
-0.3111
5.7925***
Age
Age × Age
Log total income
Log total transfers
Constant
(1.1023)
(1.4407)
(1.0809)
R2
0.15
0.17
0.12
3,048
2,462
3,155
Method: OLS.
Table 16: Age at which wealth or assets start decreasing for regressions with health or transfers
Net worth
Fin. Assets
Real Estate
with health
68
79
64
with transfers
64
74
62
31
Table 17: Ownership explained by gifts
(1)
(2)
(3)
Owner
Owner
Owner
0.0403***
0.0395***
(0.0052)
(0.0057)
0.0193**
0.0162*
0.0189**
(0.0085)
(0.0093)
(0.0088)
-0.0002***
-0.0002**
-0.0002***
(0.0001)
(0.0001)
(0.0001)
-0.0035
-0.0016
-0.0018
(0.0041)
(0.0045)
(0.0042)
-0.0769***
-0.0899***
-0.0973***
(0.0130)
(0.0143)
(0.0134)
-0.0017***
-0.0023***
(0.0006)
(0.0006)
0.0499
0.2011
0.5462*
(0.2990)
(0.3253)
(0.3037)
R2
0.04
0.05
0.03
4,022
3,472
3,914
Log total transfer
Age
Age × Age
Log total income
Gender
Age at first transfer
Constant
Source: SHARE data set, first wave. Linear probability model.
Method: OLS.
32
ownership of the house. Some of the value of the property can be ring-fenced as an inheritance for
children. Borrowers can choose to make repayments, to pay only interest, or to let the accrued
interest ‘roll-up’. The lifetime mortgage may embody several elements of insurance: caps on
the interest rate (if the rate is not fixed) and most importantly, there is a no negative equity
guarantee. According to this guarantee, when the property has been sold, if the amount is not
enough to repay the loan to the lender, the estate (i.e., the heirs or the borrower) will not be
liable for the difference between sales revenue and total debt. The latter provision entails an
element of life insurance (because of the longevity risk of the borrower) combined with insurance
of the real-estate market risks (i.e., house-price risk) — this is a key reason for the existence of
a minimum borrower age. The borrower has the right to remain in the property for life, or until
he or she needs to move to long-term care. The released money can be drawn as a lump sum or
in smaller amounts through time, or both.
Under the home reversion plan, part or all of the person’s home is sold to a specialized
intermediary in return for a lump sum or regular payments. The seller has the right to continue
living in the property until death, rent free, but agrees to maintain and insure the property.
Again a percentage of property can be ring-fenced to form a bequest. At the end of the home
reversion plan, the property is sold by the intermediary and the proceeds are shared according
to the remaining proportions of ownership. The no negative equity guarantee applies. Again,
a home reversion plan embodies an element of life insurance: the provider bears the seller’s
longevity risk and the future house price risks.
In the United States and Canada, there exists a closely related contract called the reverse
mortgage. Again, the contract allows elders to access equity built up in their homes and defer
payment until after death. Interest is added to the loan balance each month and can grow to
exceed the value of the home, but the estate is generally not required to repay any loan balance.
France has a traditional and original form of contract closely related to home reversion called
the viager contract. This form of contract is based on the breakdown of property rights in French
civil law. The property of an asset, and in particular, that of a house, can be decomposed in
three parts: usus (i.e., the right to use the good), fructus (i.e., fruit: the right to earn the asset’s
income or rents) and abusus (the right to sell or dispose of the good). The bare property is
by definition the ownership of a house, without usus and fructus, the latter being the rights to
inhabit and (or) earn the associated rents. Under the viager contract, the buyer pays a lump
sum, called bouquet, and commits to pay a sum of money every year or month, called rent,
to the seller until the seller dies. The buyer is the bare owner of the house until the seller’s
death; retrieves full ownership when the seller passes away. The seller becomes the usufructuary
(that is, is entitled to usus and fructus) when the viager is signed, for life. The buyer is clearly
bearing the seller’s longevity risk. The value of the house at the moment of the seller’s death
is also uncertain. Bare ownership means that the property is not yielding any rents. It follows
that the discounted, expected (actuarial) value of the periodical payments of the seller cannot
be greater than the discounted expected value of the future rents that can be earned after the
seller’s death. In France, most of the viager contracts are signed between individuals: until
33
recently, intermediaries played a negligible role in this market. The market is underdeveloped
because buyers must bear the longevity risk of the seller, and households cannot diversify this
risk. In addition, there is a risk of default on the part of the buyer. The volume of viager sales
could grow if powerful, deep-pocket intermediaries did invest a sufficient amount of capital in this
market, because intermediated viager contracts could render the same services as home reversion
plans. The usefulness of this type of contract may of course depend on the tax treatment of
bare ownership vs. that of usus and fructus. Bare ownership can be profitable in France because
it is exempt of the tax on capital called ISF. The home reversion plan and the French viager
are very close in economic terms. Under a home reversion contract, the seller can stay in the
home as a tenant, rent free, for life. In the viager contract the usufructuary is like a rent-free
tenant. The usufructuary can in principle let the house to earn rents, but many contracts specify
that the buyer’s full ownership is retrieved if the seller moves to another residence. Under the
latter contactual clause, home reversion and viager are almost equivalent (up to tax-treatment
differences).
3.2
Taxation of Bequests and Gifts in Europe
We will focus here on inheritance (or estate) taxes on the one hand, and on inter-vivos transfer
(or gift) taxes on the other hand. We’ll use the European Union as an example, to illustrate
the diversity of modes of taxation of bequests and gifts. Inheritance and gift taxes are taxes on
transfers of wealth. In a way they are capital taxes; this is particularly obvious when the tax
applies to real property and land, but also when a firm is transferred from on generation to the
next.
Taxation of Inheritance
Inheritance is taxed widely in Europe. To be precise, it is taxed in 18 out of the 28 EU member
states. Austria, Sweden and Portugal are notable exceptions. Inheritance taxes in Europe have
a common structure: they are levied on the market value of a net inheritance; the taxpayer is
the heir; most inheritance taxes are progressive; they discriminate according to family ties (close
relatives and spouses typically enjoy lower rates and(or) higher exemptions); the surviving child
or spouse is often fully exempt from paying the tax; there are special regimes for business asset
transfers in most national tax systems.
The treatment of close relatives has a substantial influence on the effective tax rates. Most
EU states recognize the freedom of bequeathing to close relatives to a certain extent; the most
common justification being that people should be able to take care of their relatives after death
(within limits). So, most small bequests are largely untaxed, and it should not be surprising to
find that inheritance taxes contribute a small proportion of GDP in most countries. Belgium is
the country in the EU with the highest relative importance of inheritance and gift taxation as
a share of total tax revenue; the Belgian figure is 1.37%. The equivalent figure for France, that
comes second, is 0.97%. The revenues of inheritance and gift taxes in the EU are everywhere
below 1% as a share of GDP. It is also well-known that the inheritance tax is everywhere hotly
34
debated (the “death tax”). The political economy of this type of taxation leads to a form of
instability of its provisions. Since the year 2000, five European countries have abolished the tax
completely (including Austria, Italy and Slovakia).
The tax structures in the 18 countries taxing bequests have the following common features.
Taxes are almost always general, meaning that the tax base is the entire net estate. The taxpayer
is the heir. In Denmark and the UK, the estate itself is taxed (inheritance tax vs. estate tax), but
this distinction is secondary for the purpose of the present study. The tax rate has a structure
with different scales: a different schedule applies to close relatives. The more distant is the heir,
the higher the rates. The taxes are progressive (only Denmark, Italy and Croatia use a flat rate
in the EU), but the degree of progressiveness varies substantially from one country to the other.
The lowest rates are 6% on average. For more distant heirs, the rate is 20% on average. France,
Germany, Ireland and Spain are special because the rate on children can be as high as 30%.
Third parties are in some countries taxed at very high rates (80% in Belgium and Spain). The
most common exemptions apply to small bequests and to family members: spouses and children
receive large exemptions; 11 out of 18 countries fully exempt the surviving spouse; only 7 out of
the 18 countries fully exempt the children. The transmission of family businesses poses a specific
problem. It follows that 12 out of the 18 countries taxing bequests have a business continuity
exemption, mostly in the form of a reduction in the value of the assets subject to the tax.
To compute effective tax rates, it is necessary to construct illustrative cases. We consider
three such cases: a) the modest bequest (a house worth 135,000 euros and savings equal to 33,750
euros); b) the house is worth 338,000 euros on average and the value of other assets is equal to
169,000 euro; c) the wealthy parent case, a house valued 540,000 euros, savings amounting to
135,000 and shares in a business worth 675,000 euros. We assume in turn that the heir is a
spouse (S) , a child (C), or a distant (non-related) person (D). The average effective inheritance
tax rate in the EU is 2.8% in case aC; 5.75% in case bC and 5.12% in case cC. France stands out
in Europe because of high tax rates on children with rates 7%, 15.7% and 12.3% in cases aC, bC
and cC, respectively. Denmark, the Netherlands, Belgium and Finland are similar in this respect.
The other countries have very low or zero effective rates when the heir is a child. The treatment
of spouses is different; the average effective rates are 0.89%, 1.7% and 1.7% in cases aS, bS and
cS, respectively. Finland, Belgium (Walloon Region) and Poland stand out with taxation rates
around 7%, but most EU countries have rates close or equal to zero for spouses. In contrast,
rates are on average around 20% in the (D) cases. They may be very high for non-related heirs
in countries like France or Belgium, about 60%. These figures are based on official EU data (cf.
European Commission (2014)).
Taxation of Gifts
Let us now consider gift taxes. Most EU countries have a gift tax or a provision for gifts in
the income tax. Austria, Portugal, Sweden and a number of eastern European countries (Baltic
states, Romania) are notable exceptions. Most countries have exemptions and tax the beneficiary
of gifts at a rate that depends on family relationships. Effective tax rates can be assessed using
35
the same illustrative cases as in the subsection devoted to inheritance taxes. The average effective
gift tax rate in the EU is 3% in case aC; 5% in case bC and 4% in case cC. France stands out in
Europe because of high tax rates on children with rates 7%, 15.7% and 12.3% in cases aC, bC
and cC, respectively. Denmark, the Netherlands, Belgium and Finland are similar in this respect
with effective gift tax rates around 14%. The other countries have very low or zero effective
rates when the heir is a child. The treatment of spouses is different; the average effective gift
tax rates are 2.5%, 3.5% and 3% in cases aS, bS and cS, respectively. Finland, France, Belgium
(Walloon Region) and the Netherlands stand out with gift taxation rates around 15%, but most
EU countries have rates close or equal to zero for spouses. In contrast, rates are on average
around 20% in the (D) cases. They may be very high for non-related heirs in countries like
France or Belgium, about 60%. The zero-rate countries, like Sweden, are now an exception.
Gift taxes are a small share of a country’s GDP; yet, they may be an important source of
tax revenues for some local governments and municipalities. Another important problem is tax
evasion. If the donor wants to transmit the ownership of a particular dwelling or a piece of land,
a notary will record the transaction and a gift tax can always be charged. The transmission of
shares of stock or shares of ownership in a firm may be costly to hide. The most common form
of evasion is therefore the under-valuation of the transmitted assets. But for the vast majority
of people, if the goal of parents is simply to help children, not to transmit a particular house or
a particular firm, then, parents can give money in cash to their children, without any possibility
of detection. Parents make gifts and often try to share their estate between their children during
their lifetime. It follows that inheritance and gifts are imperfect substitutes, but gifts clearly
dominate inheritance, in particular because of time preference and risk aversion.
A key question is that, insofar as they take the form of real estate, assets are illiquid for the
vast majority of the people, and it may be difficult to give part of one’s house to a child. In French
civil law, the ownership rights can be broken down in several parts: usus, fructus and abusus
(defined above) and as a consequence, the bare ownership (abusus) can be given to a child, but
this gift will be taxed. However, fiscal optimization may require that the mother sells the bare
ownership to her daughter. The latter’s gift tax will be based on the value of bare ownership only
(which is of course lower than the value of full ownership). The full ownership will be recovered
by the daughter when her mother passes away, and the inheritance of the usus-fructus part of
the mother’s house will not be taxed. The interplay of gift taxation and inheritance taxation
may therefore be rather subtle.
4
A Simple Model of Intergenerational Transmission of Wealth
We now describe and analyze a simplified model of family contracts, allowing the study of
intergenerational transmission. The basic model has three periods and three agents: the mother,
the daughter and the banker-insurer. The mother owns a house and is entitled to a stream of
income. The daughter wishes to buy a home but, possibly, she doesn’t have enough assets or
cash to buy a house that is spacious enough. The mother would like to give something to her
36
daughter: to a certain degree at least, the mother is an altruist. At the same time, the mother
wants to stay in her house; she doesn’t want to move or downsize her consumption of housing.
The banker offers to sign a contract with the mother and the daughter simultaneously. Each
of the two contracts can be seen as independent, involving the signature of the banker and one
client only, but the banker agrees to sign each of the contracts only of the other contract is signed
in the appropriate terms. Alternatively, one can view the two contracts as a single agreement
between three parties. There is no difference between the two views at least as long as contract
renegotiation is not considered. If renegotiation may happen, then of course, it is important to
specify if unilateral renegotiations are permitted (with the bank) or if the agreement of more
than two parties is needed to renegotiate. We are not considering renegotiation here.
In the simplest version of the model there is a single source of risk, namely, the risk affecting
the duration of the mother’s life. There are three periods and the mother may pass away at the
end of the first period or survive, and pass away at the end of the second period. The daughter
lives for three periods for sure. The mothers’ after-tax financial and housing wealth is entirely
transmitted as a bequest to the daughter at the beginning of the period immediately following
her death. In the simplified version of the model, the house prices, the rents and the income of
both the mother and the daughter are non-random. We focus on the life insurance problem, on
the intergenerational transfer and housing-equity release problems.
The banker-insurer offers a reverse mortgage of a special kind, or a French viager contract to
the mother and simultaneously, offers a mortgage contract to the daughter to buy her home. The
bundle of contracts simultaneously allows for (i), equity release (transforming an illiquid asset into
cash, or a sequence of cash payments); (ii), an inter vivos gift to the daughter; (iii), an ordinary
bank loan to the daughter, (iv) income annuitization for the mother; and (v), independently
of the existence of a bank loan, a certain amount of consumption smoothing through income
insurance for the daughter.
4.1
Basic assumptions
We start with a three-period, discrete-time model with contingent goods. There are three agents,
the mother, the daughter and the banker. The mother dies at the end of period t = 1 with
probability (1 − p) or survives until the end of period t = 2 with probability p. We assume
0 < p < 1.
There exists a competitive “spot” market and a rental market for housing in each period
t = 1, 2, 3, .... The spot price of a square meter in period t is denoted qt . The rent of a square
meter available in period t is denoted Rt . We assume that these prices are non-random and that
the agents have perfect foresight, for simplicity. We will see later how these assumptions can be
relaxed and how restrictive they are.
The mother owns a house of size H0 > 0. House size is expressed in units of physical area,
i.e., in square meters. The daughter buys a house of size H1 ; she may also rent square meters in
period t.
37
We now define contingent consumption. There are two goods in the economy at each period:
a consumption good and housing services. The mother’s consumption in period 1 is denoted c01 .
If the mother survives, her consumption in period 2 is denoted c02 . The consumption profile of
the daughter if the mother survives in period 2 is denoted (c11 , c12 , c13 ). If the mother dies in
period 1, the continuation consumption profile, from t = 2 on, is denoted (ĉ12 , ĉ13 ).
The daughter dies at the beginning of period t = 4; she sells her house on a futures market
in period t = 3. The daughter has no children and no bequest motive, and as a consequence, she
liquidates her housing capital entirely before death. For simplicity, the daughter doesn’t adjust
ownership after period 1: H1 is fixed in periods t = 1, 2, 3.
The daughter can also rent additional square meters at each date t = 1, 2, 3. Again we
consider contingent housing consumption. Let ht denote rented housing in square meters if the
mother survives in period 2 and let ĥt denote rented square meters in periods t = 2, 3 if the
mother passes away at the end of period 1.
The mother can sell a fraction of her property to the bank and keep the usus and fructus, that
is, the mother remains a tenant until death, paying a zero rent; i.e., she signs a French viager
contract, or a home reversion contract on a fraction of her house. Let z0 denote the fraction
of the house sold at t = 1 as a viager contract. The mother can also downsize her housing
consumption and sell a fraction zt , t = 1, 2 on the spot market in periods t = 1 and t = 2 if she
survives. We assume that zt ≥ 0 and obviously, the mother cannot sell more than her house,
0 ≤ z0 + z1 + z2 ≤ H0 .
The mother and the daughter are endowed with additively separable, instantaneous Von
Neumann-Morgenstern utility functions for consumption and housing. Let u(c) denote the instantaneous utility of consumption (the same for the mother and the daughter, to keep the model
simple). Let v0 , resp v1 denote the separable utility of housing services for the mother and the
daughter, respectively. Let β0 , resp. β1 denote the discount factors of the mother and the
daughter, respectively. The mother’s instantaneous utility for housing consumption is denoted
v0 (h0t ; H0 ), where h0t represents current consumption and H0 is the reference point, i.e., the
house owned by the mother at the beginning of period 1. We assume the following.
Assumption 1.
1. Utility v0 is continuous, strictly increasing, strictly concave and differentiable almost everywhere with respect to current housing consumption h0t . We assume the existence of a kink at
point (H0 , H0 ). The left-hand side partial derivative of v0 with respect to the first variable h0t
is higher than its right-hand derivative.
2. The utilities u and v1 are twice continuously differentiable, strictly increasing and strictly
concave functions.
The assumption relative to v0 allows us to formally represent the psychological fact that the
mother clings to her house, as a kind of “habit formation” model for housing. The concavity
assumptions formally capture risk aversion as usual.
38
Given these assumptions, the instantaneous utility of the mother in period t = 1, 2 can be
written as follows,
v0 H0 − Σtτ =1 zτ + u(c0t ).
The expected-discounted utility of the mother is defined as follows,
U = v0 (H0 − z1 ) + u(c01 ) + pβ0 [v0 (H0 − z1 − z2 ) + u(c02 )]
(1)
The instantaneous utility of the daughter is given by the following expression,
v1 (H1 + ht ) + u(c1t ).
The expected-discounted utility of the daughter can be written as follows,
V = v1 (H1 + h1 ) + u(c11 )
+
3
X
β1t−1 pv1 (H1 + ht ) + (1 − p)v1 (H1 + ĥt )
t=2
+
3
X
(2)
β1t−1 (pu(c1t ) + (1 − p)u(ĉ1t )) .
t=2
We now consider the financial contracts and the taxes.
1. A lump sum (i.e., the bouquet), denoted K is paid by the bank to the mother at the beginning
of period t = 1.
2. The mother receives an annuity from the bank in period t = 2, denoted M0 , if she survives.
3. The net transfer from the bank to the daughter is denoted Mt , t = 1, 2, 3 if the mother
survives in period 2, and denoted M̂t , t = 2, 3 if the mother passes away at the end of period 1.
4. Let xt denote the transfer from the mother to the daughter (a gift of the mother in period t
is a nonnegative xt ).
Let τg denote the tax rate on inter vivos transfers, and let τb denote the tax rate on bequests.
Let y0t , resp. y1t , denote the income of the mother, resp., of the daughter in period t.
4.2
Actuarial budget constraints
To understand how the model works, we first write the agent’s budget constraints. The mother’s
budget constraint in period t = 1 is defined as follows,
c01 + x1 ≤ K + y01 + q1 z1 .
The mother can use the bouquet K and the revenue from downsizing q1 z1 to fund consumption
and a gift x1 . The daughter’s budget constraint is
c11 + R1 h1 + q1 H1 ≤ M1 + y11 + (1 − τg )x1 .
39
We can aggregate the two budget constraints and obtain the family’s budget constraint in period
t = 1, that is,
c01 + c11 + R1 h1 + q1 H1 + τg x1 ≤ K + M1 + y1 + q1 z1 ,
where by definition, yt = y0t + y1t . The mother’s budget in period t = 2, if she survives, is as
follows,
c02 + x2 ≤ M0 + y01 + q2 z2 .
The daughter’s period-2 budget constraint, if the mother survives, is the following,
c12 + R2 h2 ≤ M2 + y12 + (1 − τg )x2 .
The aggregate family constraint in period 2 is then,
c02 + c12 + R2 h2 + τg x2 ≤ M0 + M2 + y2 + q2 z2 .
The daughter’s budget constraint in period t = 2, if the mother dies in period t = 1, is given by
the expression,
ĉ12 + R2 ĥ2 ≤ M̂2 + y12 + q2 (H0 − z0 − z1 )(1 − τb ),
where the pre-tax bequest is given by q2 (H0 − z0 − z1 ).
The daughter’s budget constraint in period t = 3 if the mother dies in period t = 1 is the
following inequality,
ĉ13 + R3 ĥ3 ≤ M̂3 + y13 +
q4 H 1
,
(1 + r)
where r is the interest rate and
q4 H1
(1 + r)
is the present value of the daughter’s house, sold at the discounted price q4 /(1 + r) but available
only in period t = 4, after the daughter’s death. We will see later the role played by the
assumption that the daughter sells her home in the last period of her life. It is in fact the
appropriate assumption in this finite horizon model. The assumption means that the daughter
can consume the value of her house in period 3 as if she benefited from a reverse mortgage at
rate r. Finally, the daughter’s budget constraint in period 3, if the mother passed away in period
2, must be expressed as follows,
c13 + R3 h3 ≤ M3 + y13 +
q4 H1
+ (1 − τb )q3 (H0 − z0 − z1 − z2 ),
(1 + r)
where
q3 (H0 − z0 − z1 − z2 )
is the pre-tax value of the bequest in this case.
We now write the banker’s intertemporal actuarial constraint. The banker can freely borrow
or lend at interest rate r. The banker invests L = M1 + K at the beginning of period 1 and
obtains a benefit B if the mother dies in t = 1 or B̂ if the mother dies in t = 2. These benefits
are expressed in present value terms, discounted back in period 1.
40
If the mother dies in period t = 2, the banker’s intertemporal budget constraint can be
written,
B−L=
(M0 + M2 )
M3
q3 z 0
− M1 − K −
−
,
2
(1 + r)
(1 + r)
(1 + r)2
where
q3 z 0
(1 + r)2
is the value of the viager sale, given that the mother survives. This is because a fraction z0 of
the mother’s property is sold by the bank at spot price q3 in period 3. If the mother dies in
period t = 1, the banker’s intertemporal budget constraint can be written,
B̂ − L =
M̂3
q2 z 0
M̂2
,
− M1 − K −
−
(1 + r)
(1 + r) (1 + r)2
where
q2 z 0
(1 + r)
is the value of the viager sale, given that the mother dies early, since a fraction z0 of the house is
sold at price q2 in period 2. We assume perfect divisibility of the mother’s property but we will
check later that this assumption is harmless because the optimal choice happens to be z0 = H0 .
Given these expressions, the expected present value of the bank’s profit is thus,
3
pB + (1 − p)B̂ − L = Xz0 −
X (pMt + (1 − p)M̂t )
pM0
− K − M1 −
,
(1 + r)
(1 + r)t−1
t=2
where by definition
q2
q3
+p
(1 + r)
(1 + r)2
is the value of a unit of the house sold under a viager contract in period 1.
X = (1 − p)
Now, the expected profit of the bank is nonnegative if and only if the following actuarial
constraint holds,
3
X (pMt + (1 − p)M̂t )
pM0
+
.
Xz0 ≥ K + M1 +
(1 + r)
(1 + r)t−1
(3)
t=2
First define the expected-discounted value of family income Y . By definition,
Y = y1 +
(py2 + (1 − p)y12 )
y13
+
.
(1 + r)
(1 + r)2
We can now substitute the mother’s and the daughter’s budget constraints, expressed as equalities, in the above nonzero profit constraint of the bank. This yields the actuarial resource
constraint, denoted (AC), in the economy under study. We easily obtain the following inequality,
c01 + c11 + R1 h1 +τg x1 + q1 H1
p
p
+
(c02 + c12 + R2 h2 + τg x2 ) +
(c13 + R3 h3 )
(1 + r)
(1 + r)2
(1 − p)
(1 − p)
+
(ĉ12 + R2 ĥ2 ) +
(ĉ13 + R3 ĥ3 )
(1 + r)
(1 + r)2
q2
(1 − τb (1 − p))
≤ Xz0 τb + (H0 − z1 − z2 )(1 − τb )X + q1 z1 + z2
(1 + r)
q4 H1
+ Y +
.
(1 + r)3
41
(4)
Remark that if the tax rate on bequests is zero, i.e., τb = 0, then the fraction of the house
sold under a viager contract is irrelevant, i.e., z0 doesn’t appear in the utility functions, and it
doesn’t appear in the resource constraint (4). The viager contract is therefore mainly used to
avoid the inheritance tax. This is because the bank and the family members can freely arrange
the profile of transfer payments in a contingent way.
4.3
Optimal family contracts
We can now study the optimal family contracts. Our main assumption is that the mother is a
pure altruist, at least to a certain extent. This is formalized as follows.
Assumption 2. The preferences of the mother over contingent resource allocation in the three
periods are represented by the utility U +γV , where γ > 0 is a parameter measuring the mother’s
degree of concern for the welfare of her daughter.
We can now formulate a key remark. Maximizing U + γV subject to the actuarial constraint
(4), the sign constraints, cjt ≥ 0, ht ≥ 0, Zt ≥ 0 and H0 ≥ z0 + z1 + z2 yields a Paretooptimal contract in the economy comprising the mother, the daughter and the banker. The set
of Pareto-optimal arrangements is described when γ varies in the set of positive real numbers.
A fully general optimization problem characterizing the Pareto-optima can be obtained if we
rewrite the resource constraint as an expression for the bank’s expected profit B̄. It is easy to
see that the bank’s expected profit is just the RHS of (4) minus the left hand side of (4). So, the
optimal contract problem can be reformulated as that of maximizing U + γV subject to V ≥ V0
and B̄ ≥ B0 , the sign constraints, and H0 ≥ z0 + z1 + z2 , where V0 and B0 are parameters of
the problem. These parameters are minimal values of the daughter’s utility and the banker’s
expected profit, resp. Varying V0 is equivalent to varying γ in the characterization exercise. In
the present section, we focus on zero-profit optima, but first order necessary conditions would
be the same if we choose a positive value of profit B̄ > 0.
Remark that the maximization of U +γV subject to (4) and sign constraints is a well-behaved
convex programming problem. The objective is a concave function and the linear constraints
clearly define a convex feasible set. It follows that the Kuhn and Tucker first-order necessary
conditions for optimality are also sufficient. These conditions are studied in the Appendix. Now,
define the left derivative of v0
v0 (h, H0 ) − v0 (H0 , H0 )
∂v0 (h, H0 ) −
.
=
lim
h→H0 ,h<H0
∂h
h − H0
The value of this left derivative at point (H0 , H0 ) represents the strength of the mother’s psychological ties to her house. It may also represent other unmodeled reasons for willing to remain
an owner occupier. The following proposition is proved in the appendix.
Proposition 1. The mother’s house is sold entirely under a viager contract if the tax rate on
inheritance is positive. If
∂v0 (H0 , H0 ) −
∂h
42
is large enough and if τb > 0, then, any optimal contract satisfies
z1∗ = z2∗ = 0
and
z0∗ = H0 ,
where starred variables denote optimal values.
For proof, see the Appendix.
We now consider the daughter’s choice of house size H1∗ and the daughter’s tenure choice. The
daughter is allowed to rent and (or) to buy a certain quantity of housing. We want to characterize
the market conditions under which the daughter indeed at least weakly prefers to buy her home
over renting. We obtain the following proposition.
Proposition 2.
1. The daughter wishes to buy her house if the following arbitrage inequality is satisfied,
q1 ≤ R1 +
R2
R3
q4
+
+
(1 + r)
(1 + r)2 (1 + r)3
(5)
2. If the daughter rents a positive quantity of housing, i.e., ĥ∗t > 0 and h∗t > 0 then, the above
arbitrage inequality holds as an equality.
3. If the arbitrage inequality holds as a strict inequality and the daughter doesn’t rent a positive
quantity of housing, i.e., ĥ∗t = 0 and h∗t = 0, then, the optimal size of her house, H1∗ , is given by
the following equation,
v10 (H1∗ )
1 + β1 +
β12
λ∗
=
γ
q1 −
q4
(1 + r)3
.
(6)
If the housing market is competitive and if the standard arbitrage equation linking rents and
prices holds (the arbitrage inequality holds as an equality), renting and buying are equivalent
decisions. In this case we could easily assume that the daughter has (at least) a slight preference
for ownership and derive the result that renting is zero. This preference for ownership may be
well-justified, for various reasons that are not captured by our simple model. A simple way of
introducing a bias towards ownership is to assume that the daughter’s utility for housing services
v1 has two arguments, i.e., we have v1 = v1 (h, H) instead of v1 = v1 (h + H), and the marginal
utility of a square meter is higher when owned. We focus on daughters with at least a slight
preference for ownership.
The next step is to study the optimal consumption profile of the mother and the daughter:
this yields indications on the consumption smoothing or income insurance properties of the
optimal family contract. We obtain the following proposition.
43
Proposition 3. The optimal consumption profile satisfies the following properties. The actuarial
resource constraint’s Lagrange multiplier is positive, that is, λ > 0, and it is equal to the marginal
utility of the mother’s consumption in the first period,
u0 (c∗01 ) = λ.
(7)
u0 (c∗01 ) = γu0 (c∗11 )
(8)
β0 u0 (c∗02 ) = γβ1 u0 (c∗12 )
(9)
We also find the sharing rules,
We find the intertemporal allocation rules,
u0 (c∗01 )
(1 + r)
u0 (c∗11 )
β1 u0 (c∗12 ) =
(1 + r)
u0 (c∗11 )
β12 u0 (c∗13 ) =
,
(1 + r)2
β0 u0 (c∗02 ) =
(10)
(11)
(12)
and the full insurance property, that is,
c∗12 = ĉ∗12
(13)
c∗13 = ĉ∗13 .
(14)
The sharing rules show how parameter γ, measuring the mother’s altruism, determines the
division of income between the two women. The intertemporal allocation rules are standard
“Euler equations”. The dynamics of consumption are determined by the discount factors β1 and
β0 . The optimal consumption is constant through time if β0 = β1 = 1/(1 + r). The last two
properties express the fact that the daughter is completely insured against the consequences of
the mother’s risk of early death. These properties imply that the daughter’s consumption profile
doesn’t depend on the event of the mother’s early death. The banker is clearly providing the
insurance here.
Finally, we determine the optimal gifts. We have the following clearcut result.
Proposition 4. If the tax rate on inter vivos gifts τg is positive, the optimal gifts are zero, that
is, x∗t = 0 for t = 1, 2.
The interpretation of the result is that, if the intermediated transfers between the mother and the
daughter are not considered as straightforward gifts, then, these transfers coming from the bank
are cheaper and should be preferred over any direct transfer of income (or property) between the
mother and the daughter. The optimal family contract may be interpreted as a hidden transfer
44
from the mother to the daughter by the tax authority. It is then possible to give a definition of
the implicit transfers made through the bank intermediary, but to a large extent, this definition
is a question of convention. We will discuss this point in greater detail below.
It is now possible, and easy, to compute the optimal family contract. The first thing that we
need is the optimal value of λ∗ . If λ∗ is known, then, using the equations stated in Proposition 3,
we derive the optimal consumption plan for the mother and the daughter. Finally, the optimal
consumption plan in turn determines the optimal bouquet K ∗ , the optimal annuity M0∗ , the
optimal net transfers between the bank and the daughter, Mt∗ and the optimal size of the
daughter’s property H1∗ .
By assumption, the utility functions u and v1 are invertible. Let φ denote the inverse of u
and ψ denote the inverse of v1 . With these definitions, it is easy to solve for optimal consumption
and housing as a function of λ∗ . We have,
c∗01
∗
c∗11
λ∗
γ
;
= φ(λ );
= φ
λ∗
λ∗
c∗02 = φ
;
c∗12 = φ
;
(1 + r)β0
γ(1 + r)β1
λ∗
∗
;
c13 = φ
γ(1 + r)2 β12
∗
λ (q1 − q4 (1 + r)−3 )
∗
.
H1 = ψ
γ(1 + β1 + β12 )
We then substitute these expressions in the actuarial resource constraint (4) above, expressed as
an equality since λ∗ > 0 implies that this constraint is always binding. This yields an equation
in λ∗ that can be solved in many standard cases. At the optimum contract, the expression for
the actuarial resource constraint boils down to,
c∗01 + c∗11 + q1 H1∗ +
pc∗02
c∗12
c∗13
H1∗
+
+
=
H
X
+
Y
+
q
.
0
4
(1 + r) (1 + r) (1 + r)2
(1 + r)3
(15)
The latter equation determines the value of λ∗ . The solution is easy to find if utilities are
homogeneous functions, for instance, with the usual CRRA form u(x) = (xa+1 − 1)/(a + 1).
Given the optimal values of c∗jt , and H1∗ we finally derive the optimal contract payments as
follows.
Proposition 5. The optimal intermediated family contract is defined by the following list of
variables.
z0∗ = H0 ;
(16)
z1∗ = z2∗ = 0;
(17)
K ∗ = c∗01 − y01 ;
(18)
M0∗ = c∗02 − y02 ;
(19)
45
M1∗ = c∗11 + q1 H1∗ − y11 ;
(20)
M2∗ = c∗12 − y12 = M̂2∗ ;
(21)
M3∗ = c∗13 − y13 −
q4 H1∗
= M̂3∗ .
(1 + r)
(22)
The optimal contract is characterized by the absence of inheritance, full housing equity release
on the part of the mother, full annuitization of the mother’s income, consumption smoothing for
the daughter. The optimal contract typically entails a loan to the daughter, so that we expect
M1∗ > 0 because the period 1 transfer to the daughter covers the price of the house q1 H1∗ and later
transfers may be negative to repay the loan, principal and interest, that is, we would typically
have M2∗ < 0 and M3∗ < 0.
5
Extensions and Generalizations
We can now study various kinds of extensions and generalizations of the basic model. There are
several kinds of interesting extensions.
1. Multi-period extension of the basic model with a single heir.
2. Multi-person extensions of the model to accommodate several children.
3. Multiple-risks extensions of the basic model. The basic model treats the problem of life
insurance. But other sources of risk are important, mainly:
(a) The risk of a severe health problem, forcing the mother to move to a nursing or
retirement home, and requiring additional resources (long-term care risk).
(b) The income risk borne by the daughter. The daughter may become unable to face
mortgage repayments, leading to foreclosure.
(c) The risks affecting property prices and rents.
4. The banking monopoly model.
The problem may become difficult to analyze in its full generality, but we know that the four
kinds of extensions listed above are feasible, using the same basic economic principles, at the
cost of some simplifying assumptions, to keep the computations tractable. Point 4 in the above
list boils down to considering profit maximization instead of utility maximization. The optimal
contracts that maximize the banker’s expected profit are easy to characterize: they basically
share the same features of all optima, but a complete study of the bank monopoly problem
requires a careful description of the family’s individual rationality (or participation) constraint.
We start with a sketch of the multiperiod extension.
46
5.1
Multiperiod Extension
Let the time horizon of the model be denoted T + 1. The daughter dies at the end of this period.
The mother dies at the end of period T , which is random. We assume that 1 ≤ T ≤ T . The
daughter will outlive her mother by at least one period. Let pT be the unconditional probability
of the mother dying at T . Thus, we have,
T
X
pT = 1.
T =1
The cumulative distribution is denoted Pt , that is, by definition,
Pr(T < t) =
t−1
X
pT = Pt .
T =1
The survival function is denoted St . By definition again, we have,
St = Pr(T ≥ t) = 1 − Pt .
The conditional density, or hazard rate, is therefore,
Pr(T = t | T ≥ t) =
pt
.
St
To pose the optimal-contract problem in a fully general and rigorous way, we consider again
contingent goods. The family contract becomes contingent with respect to the mother’s death
event, i.e., to the drawing of T ∈ {1, ..., T }.
Let (c0t , c1t ) denote the consumption allocation if the mother survives in period t. Let H1 be
the size of the daughter’s house in square meters and let ht be the rented housing, also in square
meters, if the mother survives in period t.
If t > T , let c1t (T ) denote the contingent consumption of the daughter in periods following
the mother’s death at T . Additional contingent rented space is denoted ht (T ) for t > T .
The mother can still decide to sell a fraction z0 of her house under a viager contract and she
can still downsize her housing consumption at any period t ≤ T . Let then zt , t > 0 denote the
fraction of the mother’s property sold on the spot market at date t. We have the sign constraints
zt ≥ 0 for all t and the constraint,
H0 ≥
T
X
zt .
t=0
The mother’s utility, conditional on death at date T , denoted UT , can be written
"
!
#
T
t
X
X
UT =
β0t−1 v0 H0 −
zθ + u(c0t ) .
t=1
(23)
θ=1
The mother’s expected utility is simply
U=
T
X
T =1
pT UT =
T
X
T =1
47
pT
T
X
t=1
β0t−1 Ut ,
(24)
where, by definition, Ut = v0 (H0 − Σtθ=1 zθ ) + u(c0t ). We can easily rewrite this expression as
follows,
U=
T
X
β0t−1 Ut
t=1
T
X
pT ,
T =t
and therefore we finally obtain,
U=
T
X
β0t−1 St v0 (H0 − Σtθ=1 zθ ) + u(c0t ) .
(25)
t=1
The daughter’s utility, conditional on T is denoted VT , and by definition, we have,
VT =
T
X
β1t−1 [v1 (H1 + ht ) + u(c1t )] +
t=1
TX
+1
β1t−1 [v1 (H1 + ht (T )) + u(c1t (T ))] ,
(26)
t=T +1
and
V =
T
X
pT VT .
(27)
T =1
Using the fact that ST +1 = 0 we can rewrite V as follows,
V =
TX
+1
β1t−1 St [v1 (H1 + ht ) + u(c1t )]
t=1
+
T
X
T =1
pT
TX
+1
(28)
β1t−1 [v1 (H1
+ ht (T )) + u(c1t (T ))] .
t=T +1
We assume as above that the mother is a pure altruist and wishes to maximize U + γV , where
γ > 0 is a parameter measuring the degree of altruism.
We now consider budget constraints and derive the actuarial resource constraint in its general
form. Let (Mjt ) denote the transfers from the banker to the mother (j = 0) and to the daughter
(j = 1). The transfer to the mother stops after death, that is, M0t = 0 if t > T . The lump
sum called bouquet above is just K = M01 here. Given these definitions, the mother’s budget
constraints are written as follows,
c0t + xt ≤ M0t + y0t + qt zt ,
(29)
for all t ≤ T . The daughter’s budget constraints, for t ≤ T , are written as follows,
c1t + Rt ht + q1 H1 δ1t ≤ M1t + y1t + (1 − τg )xt ,
(30)
where δ1t = 1 if t = 1 and 0 otherwise. The family’s aggregate budget constraint now becomes,
c0t + c1t + Rt ht + q1 H1 δ1t + τg xt ≤ Mt + yt + qt zt ,
again for t ≤ T . To simplify notation, we denote
Mt = M0t + M1t
and
48
yt = y0t + y1t .
(31)
After T , the bank’s transfers to the daughter at t are contingent to T , denoted Mt (T ). The
daughter’s budget constraints have a different structure for t > T . If t > T + 1 and t < T + 1,
we have,
c1t + Rt ht (T ) ≤ Mt (T ) + y1t .
(32)
At time t = T + 1 the daughter receives the mother’s bequest and pays the inheritance tax. We
have,
c1,T +1 (T ) + RT +1 hT +1 (T ) ≤ MT +1 (T ) + y1,T +1 + (1 − τb )qT +1 H0 − ΣTt=0 zt .
(33)
At time t = T + 1 the daughter’s house is sold at price qT +2 /(1 + r) and the daughter dies at
the end of period T + 1. If T < T we have the final budget constraint,
c1,T +1 (T ) + RT +1 hT +1 (T ) ≤ MT +1 (T ) + y1,T +1 + qT +2
H1
.
(1 + r)
(34)
The banker’s expected profit, conditional on T , is denoted πT . For 1 ≤ T ≤ T , we have,
πT =
T
TX
+1
X
Mt
qT +1 z0
Mt (T )
−
−
.
T
−1
t−1
(1 + r)
(1 + r)
(1 + r)t−1
t=1
(35)
t=T +1
The bank’s expected profit is by definition,
X
π=
pT πT .
T
The bank’s profitability constraint is simply
π ≥ 0.
This can be rewritten,
π = z0 X −
T
X
t=1
St
T
TX
+1
X
Mt
Mt (T )
−
p
≥ 0,
T
t−1
(1 + r)
(1 + r)t−1
t=1
(36)
t=T +1
where the unit value of a viager contract is given by the following expression,
T
X
pT qT +1
X=
.
(1 + r)T
(37)
T =1
To simplify notation, define the discount factor
β=
1
.
1+r
Now, if we substitute the agent’s budget constraints, expressed as equalities, in the profitability
constraint π ≥ 0, taking into account the fact that S1 = 1 and ST +1 = 0, some easy algebra
yields the following actuarial resource constraint,
q1 H 1 +
T
X
β
t−1
St (ct + Rt ht + τg xt ) +
t=1
T
X
T =1
≤ z0 X + Y +
T
X
pT
TX
+1
β t−1 (c1t (T )+Rt ht (T ))
t=T +1
β t−1 St qt zt + (1 − τb )
t=1
T
X
β T pT qT +1 H0 − ΣTt=0 zt + qT +2 β T +1 H1 .
T =1
(38)
49
The left-hand side of the 38 is the sum of the values of all contingent goods, including gifts, the
price of which depends on the tax rate τg The right-hand side of 38 is the sum of all resources
in expected present value terms. More precisely
1. The first term is the value of the viager sale z0 X.
2. The second term is the family’s expected wealth, derived from income flows, that is, by
definition,
Y =
TX
+1
β t−1 (St yt + Pt y1t ).
t=1
3. The third term on the RHS, that is, ΣTt=1 β t−1 St qt zt , is the expected value of revenues
related to planned property-downsizing sales.
4. The next to last term is the net expected after-tax value of inheritance,
(1 − τb )
T
X
β T pT qT +1 H0 − ΣTt=0 zt .
T =1
5. The last term is the value of the daughter’s house once resold after the daughter’s death.
It follows that the effective present price of property is
q1 − qT +2 β T +1 .
It is now possible to study the optimal family contract in the multiperiod extension of our
model. The results obtained are, in essence, the same as before. The optimal intermediated
family contracts are obtained as solutions of the following program: maximize U + γV subject
to the actuarial resource constraint (hereafter ARC), 38.
Let λ denote the Lagrange multiplier of the ARC. Let µ denote the Lagrange multiplier of
the feasibility constraint H0 − ΣTt=0 zt ≥ 0. Then, from the first-order necessary conditions for
optimality, we find the equation,
λτb
T
X
β T pT qT +1 = µ.
(39)
T =1
Exactly as in the simplified version of the model, we can state a number of results characterizing
the optimal contract. Proposition 6 gathers a number of claims, the proof of which is in the
Appendix. We find that if the mother’s marginal utility of housing is large enough, she will
sell 100% of her house under a viager contract. The conditions under which the daughter
prefers ownership at period t = 1 over renting are, in essence, the same as before. The optimal
consumption satisfies properties similar to those stated as Proposition 5 above.
Proposition 6.
(i) If the tax rate on bequests τb , is positive, and if the mother’s marginal utility of housing, v00 ,
is large enough at reference point H0 , then, any optimal arrangement satisfies,
z0∗ = H0
and
zt∗ = 0 all t > 0.
50
(ii) The daughter prefers ownership at t = 1 over renting during all subsequent periods if the
following arbitrage inequality holds,
q1 ≤
TX
+1
β t−1 Rt + β T +1 qT +2 .
t=1
(iii) Assuming that the arbitrage inequality holds and q1 > qT +2 β T +1 (the first period price of a
square meter is not too low with respect to the fundamental discounted value of a square meter),
then, the daughter’s house size is uniquely determined by the following equation,
λ∗ (1 − β1 )
q1 − qT +2 β T +1 .
T
+1
γ (1 − β1 )
v10 (H1∗ ) =
(40)
(iv) The optimal consumption path is determined as follows.
1. The Lagrange multiplier λ∗ is positive and
u0 (c∗01 ) = λ∗ .
2. The sharing rule is determined by γ as follows,
γu0 (c∗11 ) = u0 (c∗01 ).
3. The inter-temporal allocation of consumption is determined by the Euler equations,
0
u
(c∗jt )
0
=u
(c∗j1 )
β
βj
t−1
,
for all j = 0, 1 and all t ≤ j(T + 1) + (1 − j)T .
4. The contingent consumption c1t (T ) doesn’t depend on pT , provided that it is positive:
we find u0 (c∗1t (T )) = (λ/γ)(β/β1 )t−1 and therefore, given the Euler equation above, the full
insurance property holds, that is,
c∗1t (T ) = c∗1t
for all t > T .
5. Finally, optimal gifts are zero because there is a tax on gifts. Kuhn-Tucker conditions
yield λ∗ β t−1 St τg x∗t = 0. It follows that λ∗ τg > 0 implies x∗t = 0 for all t ≤ T .
To solve the model completely, we again need to compute the optimal value of λ. This can
be done by first solving for the consumption profile as a function of λ. Then, the ARC provides
an equation that determines λ∗ — the entire solution depends on this key parameter. This
equation is nonlinear and must be solved numerically, except in special cases. Going back to the
period-by-period budget constraints for the mother and the daughter taken separately, we derive
∗ values.
the optimal values of transfers with the bank, i.e., the Mjt
The multi-period extension of our family contract model provides a tractable benchmark to
study optimal arrangements with the bank.
51
5.2
Multi-person Extension
Our model can easily be adapted to study the case of several children, several heirs. Assume
for instance that the mother has k children, k > 1. Let γi > 0 the weight of child i in the
mother’s utility function; let Vi be the expected utility of child i. The mother is altruist and
P
wishes to maximize U + i γi Vi . Again, any maximum of this objective function subject to the
bank’s zero-profit constraint is a Pareto-optimum. The multi-person extension doesn’t pose any
particular problems. Using additional constraints, like equal-sharing constraints, any restrictions
bearing on the children’s shares of the mother’s estate can be embodied: the model can take the
legal inheritance rules into account.
5.3
Multiple-risks Extensions
The various sources of risk lead to further generalizations of the family contract model. We
discuss in turn the rent and house-price risks; the risk of a severe health problem; the risks
affecting income.
1. Variability of rents and house prices.
Until this point we assumed that (qt )t and (Rt )t were perfectly predictable sequences. Given that
we focus on solutions in which the mother and the children are owner-occupiers, the variability of
rents doesn’t play a direct rôle in the derivation of the solution. This variability, of course, would
come into play if we studied a family model with variable tenure choice. Our benchmark contract
includes a full viager sale of the mother’s house, meaning that the mother is fully protected (by
the bank) against fluctuations in the value of the estate. Thus, the variability of rents and house
prices is mainly a problem for the bank. If we go back to 38 above, we see that the key variable
is the assessment of the viager sale XH0 . Recall that X is the discounted and expected value
of a square meter under a viager contract with the mother. To be more precise, with almost no
other change in the model, X could be replaced with a more general definition,
X = ET β T Eq (qT +1 | T, q0 ) ,
where E is the mathematical expectation operator. The outer operation, i.e., ET , requires only
the knowledge of life tables and the age of the mother. This is routine work for actuaries. The
other operation, that is, Eq (. | T, q0 ) is a more difficult econometric problem, a real-estate priceforecasting problem. There are models, estimation methods and data sources to perform this job.
But like other asset-price forecasting models, the variance of predictions increases with the time
horizon, and the length of the phase during which the bank pays an annuity to the mother may
easily be longer than 10 years. It follows that we have a real technical difficulty for the bank. But
as asset-value risks in general, they can be diversified away or shared, at least partially, with the
help of financial markets and the cooperation of specialized financial partners. A well-developed
market for the type of contracts under consideration provides a sequence of discount rates, taking
the risk affecting future house prices into account. We conclude that the pricing formula for X
(i.e., (37)) can be replaced with a more sophisticated or more realistic expression.
52
More generally, the mother may wish to stay an owner-occupier because, (i), owning her house
is like a call option with a given exercise price on a given number of square meters in the future;
and because, (ii), owning a house also immunizes the owner-occupier against the variability of
rents; finally, (iii) there may be various tax optimization, country-dependent reasons for which
owning one’s home is advantageous.
(i) Ownership doesn’t insure the landlord against fluctuations in property prices, but directly
secures access to a number of square meters, in a given city; thus, it secures access to a quantity
of real housing service consumption. In addition, in spite of being relatively illiquid, homeownership is at the same time a relatively flexible store of value that can be used in case of an
accident or of severe health problems: downsizing can in these cases be a way of releasing cash to
cover unexpected expenses. So, the value of home-ownership is not independent from the quality
of the mother’s health insurance (see below). Ownership is also flexible because, to a certain
extent, it allows for mobility in geographical space: the mother can move and keep more or less
the same consumption of housing services.
(ii) In a well-arbitraged property market, prices, rents and interest rates are related but the
risk affecting rents can be considered in isolation. The variability of house prices depends on the
variability of rents, interest rate shocks and irrational exuberance, that is, the possible emergence
of a real estate bubble. Renters are exposed to the variability of rents, and this variability
translates into a variability of real housing consumption as well as ordinary consumption, through
the household’s budget constraint. An owner-occupier (not a pure landlord) is typically insured
against these risks, even if he(she) plans to move under some contingencies.
(iii) The tax regime may also change the motives for home-ownership. As compared to a
renter, an owner-occupier typically pays a property tax (or a tax on real estate capital), but
since imputed rents are not taxed, with equal incomes, the renter and the owner pay the same
income tax. The owner-occupier may sometimes in addition deduct mortgage interest payments
from taxable income. It is well-known that in some countries, housing policies strongly favor
ownership in this manner. Note that a viager contract, insofar as it transfers property to the
banker, leads to a change in the mother’s wealth and real-estate capital: these changes may
trigger some additional advantages, like avoiding the property and wealth taxes.
2. Risk of severe health problems. Long-term care insurance.
The risks affecting health are of course crucial in old age. It may be that the reason why the take
up rate of equity release products such as reverse mortgages and annuities is low because seniors
are not well insured against these risks. In particular, the risk of an accident generating the need
for costly assistance at home or the need to move to a retirement (or nursing) home, must be
taken into consideration. The seniors would not want to sell their home under a viager contract
or release equity because their house is a store of value, a precautionary saving against this type
of need. Typically, the mother’s apartment could be rented to generate the additional income
needed to pay for a nursing home. Full property would typically be transferred to the buyer (the
bank) under a typical viager contract, when the mother leaves her house, for whatever reason.
53
This is true, except if the viager contract is specified differently: it would clearly be possible to
add covenants according to which the mother can let her apartment to generate cash in case of
departure to a nursing home. This possibility could explicitly be open until she passes away.
Alternatively, the banker could provide an insurance against this type of health risk in addition
to the viager contract. The bundle of liquidity, credit and insurance services offered to the family
can clearly be extended to include some long-term care insurance.
As a first approximation, we can simply model these risks as an income shock (the nurse’s
salary has to be paid out of current income). The simplest possible extension of the three-period
model presented above is to assume that if the mother survives in period 2, she also suffers
an income shock, with a certain probability, before passing away at the end of period 2. The
insurance against this risk would take the form of an adjustment of M0 , the annuity payment,
contingent on survival. This payment would simply become contingent on survival and loss of
autonomy. This risk can be priced easily and the actuarial resource constraint can be adjusted
to take it into account. Thus, a multi-period extension of the model with an additional risk of
loss of autonomy, taking the form of an income shock, poses no special technical problems.
3. Income shocks and mortgage foreclosure.
We now discuss the random income shocks affecting the daughter(s). In case of a drop in current
income, the daughter may no longer be able to repay her debt to the bank. Because of severe
moral hazard and adverse selection problems, temporary or permanent losses of income cannot
be insured by the market. Yet the loans of a certain size, granted to certain categories of clients
are insured, but only against disability and the occurrence of premature death. It is well-known
that most ordinary home loans are secured by collateral, typically by the property itself. It is
therefore reasonable to extend the model to capture the event of foreclosure. We can simply
assume that the daughter incurs a severe loss of income with some probability at each period,
and that in that event, the daughter’s house is seized and sold by the bank. Again, this type of
extension can be treated by the introduction of additional states of nature, by adjustments of the
actuarial resource constraint, to deal with new events in which the mother survives (or not) and
simultaneously, the daughter loses her income (or not), etc. Foreclosure has a present expected
value appearing as an additional resource for the bank, while the daughter’s net repayments to
the bank must be discounted for the appropriate probability of repayment. This type of extension
will not change the optimal solution in a radical way.
5.4
Banking Monopoly Model
The basic family contract model can be turned “upside down” to study surplus extraction by
the banker. Instead of maximizing U + γV subject to, say π ≥ 0, or π ≥ πmin we study the
profit maximization problem, that is, maximize the expected profit π, subject to U + γV ≥ W0 ,
where W0 is the best outside option of the family. By definition, the best outside option of the
family is what the family can achieve without any help from the bank, when the mother and
the daughter are subjected to taxes, subjected to all budget constraints, if the only financial
54
products that they can use are a savings account with rate r and an ordinary mortgage with rate
ρ and duration d = T + 1 and if they cannot borrow otherwise.
We first define what the daughter alone can achieve if she can only save money at rate r, and
cannot borrow, apart from obtaining an ordinary mortgage loan with rate ρ > r and duration d,
and there are no gifts of the mother, i.e., xt = 0 for all t. Let V0 (z1 , ..., zT ) denote this expected
utility level. This is a function of (zt ), t > 0, since the daughter receives the mother’s bequest
and since this bequest, in turn, is determined by T and the sequence (zt ).
By definition, V0 is the maximum of V subject to these constraints. Under these constraints,
the daughter cannot insure herself against the risk of receiving the mother’s bequest late, although
she can borrow to buy a house under an ordinary mortgage contract, knowing that inheritance
will help with the reimbursement of the loan at some random point in the future.
Given that V0 (z) is known, the individual rationality level W0 of the mother can be defined
as the value of the following program:
Maximize U + γV
subject to all the mother’s and the daughter’s budget constraints, subject to V ≥ V0 , allowing
for a savings account at rate r; no equity release; no borrowing, apart in the form of an ordinary
mortgage; non zero transfers xt at each date are permitted, but K = Mt = Mt (T ) = 0.
We could of course consider variants of these problems, depending on the feasible sets of
actions available to the mother and the daughter when they stand alone. The value created by
the bank is due to equity release and the various kinds of insurance offered to the mother and
the daughter, plus the tax savings, potentially. This surplus can be extracted by the bank. The
bank can also exploit the fact that the mother has a high marginal value for housing consumption
at the reference point H0 . We study the banking monopoly model in greater detail in the next
section.
6
Housing, Welfare and Profitability: Calibration and Numerical
Simulations
We now endeavour to study the model by means of micro-simulations. We first specify the
model fully by choosing functional forms for utility functions. We then calibrate the model,
using appropriate actuarial life tables and values of parameters that are viewed as acceptable in
the literature on risk aversion. Next, we generate a random sample of families reflecting these
parameters. We compute the optimal contract for each individual in the sample and compute
the value of an outside option for each of these families as well. We compute the utilities and the
profit of a bank underwriting the family contract. Finally, we study the properties of housing
demand, consumption, expected utility and expected profit in the artificially generated sample
of families.
The optimal contract under study is the full-insurance family contract, exactly as defined in
sub-section 5.1 above, with two-members families (a mother and a daughter). A more delicate
55
choice is the outside option, because there are many possible variants of this option. The results,
and in particular the bank’s expected profit per contract depend on the next best opportunity
of the family. We could for instance assume that the outside option has all the consumption
smoothing possibilities of the optimal contract but no insurance against the mother’s longevity
risk. This type of outside option can be clearly defined analytically but the numerical simulation
problems posed by the study of this form of solution are complicated. We decided to study a
less sophisticated, but equally appealing form of outside option.
6.1
Definition of the Outside Option: the Rule-of-Thumb Contract
The rule-of-thumb (hereafter, the RoT) contract is an arrangement with the bank defined as
follows. 1◦ The daughter buys a house of size H1 at date t = 1 with a mortgage from the bank
at a rate ρ > r. The mortgage is a constant amortization loan that lasts during the daughter’s
entire life, for simplicity. 2◦ The daughter sells the house during the last period of her life and
gets a constant annuity (at rate r) during her entire life as a consequence. This is to make sure
that the total expected and discounted value available to the family as a whole is the same in the
RoT and in the optimal contracts. 3◦ The daughter receives the bequest at date T + 1. Again,
the mothers’ house (of size H0 ) is sold immediately after the mother’s death and transformed by
the bank into an annuity (at rate r) that pays a constant sum until the daughter passes away.
This last assumption ensures a form of consumption smoothing that is not perfect, since the rate
of consumption of the daughter, under the RoT contract, has an upward jump at date T + 1, due
to inheritance. 4◦ A rule-of-thumb is used by the banker to determine the size of the daughter’s
house, and therefore, the amount of the loan. The mortgage repayment must be equal to a
constant fraction φ (say, φ = 1/4) of the daughter’s income y1 . This assumption entails the fact
that in the RoT contract, the banker doesn’t take the mother’s income and wealth into account
to determine the loan size. 5◦ . Finally, there are no gifts from the mother to the daughter. In
the outside option formulation, the daughter does not save: she just reimburses the mortgage
and consumes the rest of her income. For simplicity, there is no downpayment to buy the house
(but a nonzero downpayment of the daughter could easily be added).
We summarize the above points formally as follows.
a) The daughter borrows to buy a house of size H1 at rate ρ > r. Define the discount factor
δ=
1
,
1+ρ
and recall that β = 1/(1 + r). The daughter’s mortgage is repayed during the daughter’s entire
life with a constant amortizement annuity AH1 , where
A=
q1 (1 − δ)
.
(1 − δ T +1 )
b) The daughter resells the house on a futures market in the year T + 1 at price βH0 qT +2 .
The constant annuity with the same discounted value is denoted A1 H1 where
A1 =
qT +2 β T +1 (1 − β)
.
(1 − β T +1 )
56
c) The after-tax annuity value of the bequest, qT +1 H0 (1 − τb ), paid between periods T + 1
and T + 1, has the following expression,
A0t (T )H0 = (1 − τb )H0
qT +1 (1 − β)
,
(1 − β T −T +1 )
for periods t > T and conventionally we set A0t (T ) = 0 if t ≤ T .
We can now define cRoT
(T ), the daughter’s consumption at date t, under the RoT contract.
t
The income is contingent on T , since the mother passes away at the end of this period. We have,
for all t,
cRoT
1t (T ) = y1t + (A1 − A)H1 + A0t (T )H0 ,
where A0t = 0 for t ≤ T .
d) In each period t ≤ T , the mother consumes c0t = y0 . After the mother’s death (t > T ),
we simply set c0t (T ) = 0.
e) Finally, we apply the rule of thumb to determine H1 . That is, we choose a fraction φ in
the interval [0, 1] and we set H1 so that AH1 = φy1 .
6.2
Specification and Calibration
We need to specify the utility functions u and v1 and v0 . The crucial functions are in fact only
u and v1 , since we explore solutions in which the mother’s house is sold only when the mother’s
passes away. We choose standard CRRA utilities, namely,
u(c) =
c1−σ − 1
,
1−σ
if σ 6= 1, where the index of relative risk aversion is σ ≥ 0, and
v1 (H) =
H 1−η − 1
,
1−η
if η 6= 1, where 1/η determines the daughter’s elasticity of housing demand. If σ = 1 we take
the limit function u(c) = ln(c). Similarly, if η = 1 we take v1 (H) = ln(H).
In addition, we assume that house prices are predictable with a rate of growth κ, that is,
qt = q1 (1 + κ)t−1 .
The key parameters of the model are (β0 , β1 , σ, η, γ, κ, r), the psychological discount factors,
the degree of risk aversion, the inverse elasticity of housing demand, the degree of altruism of
the mother, the (real) rate of growth of house prices and the base rate of interest.
To calibrate the model completely, we need to specify the joint probability distribution of
income and housing assets, that is the joint distribution of (H0 , y0 , y1 ). We assume that these
variables are jointly log-normal, i.e., that (ln(H0 ), ln(y0 ), ln(y1 )) is normally distributed. To
determine the covariance of ln(y0 ) and ln(y1 ), we use the literature on intergenerational transmission of earnings and wealth. Lefranc and Trannoy (2005) have estimated the relation between
the daughter’s and the mother’s income with the help of French data. The standard approach
to these questions is to specify a simple econometric model of the form,
ln(y1 ) = a0 + a ln(y0 ) + ,
57
where is an independent normal error term. If ln(y1 ) and ln(y0 ) have the same variance, then,
the regression coefficient a is equal to the correlation coefficient of the two logarithms. According
to Lefranc and Trannoy (2005), a is somewhere between 0.25 and 0.45, closer to 0.4 in the recent
years. We choose a = 0.4, and equal variances for the log-incomes. The variance of ln(y1 ) is
taken from INSEE’s Enquête Patrimoine and we find that the standard deviation of log-income
is 0.83. We assume V ar(ln(y1 )) = V ar(ln(y0 )) = 0.832 = 0.69. From this we derive,
V ar() = V ar(ln(y1 ))(1 − a2 ) = 0.69 × 0.84 ' 0.58.
The standard deviation of is therefore
√
0.58 = 0.76. From Enquête Patrimoine we also get
E(ln(y0 )) = 11. We assume E(ln(y0 )) = E(ln(y1 )) for simplicity (this corresponds to a stationary
environment). This assumption pins down a0 , we have
a0 = (1 − a)E(ln(y1 )) ' 6.6.
We specify the distribution of H0 in the same way, using French data from INSEE’s Enquête
Patrimoine. We use a regression of the form
ln(H0 ) = b0 + b ln(y0 ) + ξ,
where ξ is an independent normal random error term, to model the correlation of H0 and y0 . The
regression results obtained are precise. We obtain the following estimates, b = 0.25, b0 = 1.88
and the standard deviation of ξ is equal to 0.39.
Given these specifications, we draw a random sample of 1000 families in the joint log-normal
distribution so obtained, where the variances of ln(H0 ) and ln(y0 ) are chosen in accordance with
data sources.
There are N = 1000 families indexed by i = 1, . . . , N , and we draw different ages above 50
for the mother at time t = 0. The mother’s age is drawn uniformly between 50 and 70. From this
we derive the maximal number of years of mother i and of daughter i, assuming that all mothers
pass away before the age of 100. We also use the appropriate INED life tables to compute the
survival function STi and the probability piT for each mother i in the sample.
We can now treat the simulated data by means of a statistical software as if it was a sample
of real observations, and we can generate as many samples as we wish by varying the values of
the model’s structural parameters, to perform sensitivity analysis.
Given this sample of families, we solve the model, for each of the 1000 families in the sample,
in 10 different scenarios. These scenarios are different combinations of key parameters. We study
the effect of changing three parameters only: risk aversion, altruism and time preference. There
is a ‘central scenario’ based on the choice (σ, γ, β0 ) = (.8, 1, .96) and we take r = .03, ρ = .04,
q1 = 5250 (in euros per square meter) with a real rate of growth of 1 percent per year, β1 = β0 ,
and η = 0.9. We then construct the other scenarios by changing the values of each parameter
separately while the others are fixed at their central values. The values of σ are chosen in the
set {.5, .6, .7, .8, .9}; the values of γ vary in the set {1, 1.5, 2}, and the values of β0 are chosen
in the set {.95, .96, .97, .98}. Figure 20 shows the empirical distributions of the key exogenous
58
Figure 20: Distribution of log-income for mother and daughter, and distribution of mother’s
0
0
.2
.002
.4
.004
Density
.006
Density
.6
.008
.8
.01
1
house size
100
200
Size of the mother's house
300
400
3.5
4
4.5
5
Log size of the mother's house
5.5
6
0
0
.1
.1
.2
.2
Density
.3
Density
.3
.4
.4
.5
.5
0
8
10
12
Log income of the daughter
14
9
10
11
12
Log income of the mother
13
Source: Simulated data set. Key exogenous variables
variables in the simulated sample. Table 18 gives the summary statistics. The figures show that
simulated observations represent relatively well-to-do individuals from the Paris region. Real
estate prices q1 are chosen as an average of the Paris region (average prices in the center of Paris
would be closer to 8000 euros per square meter). Given the basic data describing each randomly
drawn family, we compute the optimal, full-insurance contract as described in sub-section 5.1
above. We compute the numerical values of utilities for each family member and each family i,
that is, Ui (π), Vi (π), where π ≥ 0 is the bank’s expected profit from the contract. The utility
levels depend on the profit cashed in by the bank on each family: πi is a summary in present
actuarial value terms of the surplus extracted by the bank from family i. The appropriate value
of πi depends on family characteristics. Let πi∗ denote the maximal profit of the bank with the
Table 18: Summary statistics of simulated data
Mean
Median
Standard deviation
Income of the mother
85,759
59,586
81,440
Income of the daughter
78,143
60,193
69,222
113
102
51
Size of the mother’s house
Simulated data set. Yearly incomes are in euros. Areas are in square meters.
59
Table 19: Summary statistics of simulated profits
Mean
Median
Standard deviation
π̂(ρ) (RoT)
63,545
48,554
59,467
π∗
387,320
255,000
454,574
(CwP)
Simulated data set, central scenario. Profits are in
euros. RoT refers to rule-of-thumb contract. CwP
refers to the optimal contract with maximal profit.
optimal contract. By definition, πi∗ is such that the family is indifferent between the optimal and
the RoT contracts. Let Ui0 , Vi0 and Wi0 = Ui0 + γVi0 denote the utility levels of the mother,
the daughter and the family, respectively, under the RoT contract. Then, by definition, πi∗ is the
value of profit that solves the equation,
Ui (πi∗ ) + γVi (πi∗ ) = Wi0 .
(41)
Due to the monotonicity properties of indirect utility, this solution is the maximal value of profit
generated by a contract with family i. Our simulation program computes this value for each
family in the sample. We end up with a data set including exogenous data (Hi0 , yi0 , yi1 ), and
endogenous data
(Hi1 , Ui (πi∗ ), Vi (πi∗ ), Ui0 , Vi0 , Ĥi1 , πi∗ , π̂i (ρ))
where Ĥi1 is the daughter’s house size and π̂i (ρ) is the profit of the bank under the RoT contract.
The latter value of profit is simply,
π̂i (ρ) = φyi1
t=T
X+1
β t−1 − q1 Ĥi1 = φyi1
t=1
t=T
X+1
(β t−1 − δ t−1 ) > 0.
(42)
t=1
In particular, an output of interest is the distribution of expected profits πi∗ generated by the
profit-maximizing family contracts. Table 19 gives the main descriptive statistics of the simulated
profits. Profit is nonnegative and substantial. The reader must keep in mind that the profit,
expressed in euros, is the total discounted sum of intertemporal expected benefits, earned by the
bank over the entire life of a contract with one family, and that the chosen families are (relatively)
rich Parisians. Figure 21 shows the distribution of ln(πi∗ ) and ln(π̂i (ρ)). It is clear from Table 19
and Fig. 21 that the bank’s profit increases very significantly as compared to the RoT-contract
profit. The RoT (i.e., Rule of thumb) and CwP (i.e., Contract with maximal profit) cases can
be interpreted as providing a lower and an upper bound on the profit of the bank per capita.
Surplus sharing may lead, according to these simulations, to a fourfold or fivefold increase of
expected profits per family.
To check how the model works, we ran the artificial OLS regression of log-profit on mother’s
and daughter’s log-income and log-size of the mother’s house. Results are given in Table 20.
The first column shows that the RoT profit doesn’t depend on the mother’s wealth and income.
In contrast, all covariates become highly significant under the optimal contract, because profit
then depends on the mother’s income and wealth.
60
0
.2
.4
.6
Figure 21: Density of simulated log-profits with RoT and optimal contracts
8
10
12
x
Profit (RoT)
14
16
Profit (CwP)
Source: Simulated data set, central scenario. RoT refers to Rule of
thumb contract. CwP (‘contract with profit’) refers to the optimal
contract with maximal profit π ∗ .
Table 20: Regression of log-profit on log-incomes and log-size of the mother’s house
Log-Profit (RoT contract)
Log-Profit (Optimal contract)
-0.001
0.158***
(0.001)
(0.031)
0.000
0.535***
(0.001)
(0.055)
1.001***
0.151***
(0.001)
(0.030)
-0.030***
-0.048***
(0.000)
(0.004)
1.610***
9.563***
(0.010)
(0.403)
R2
1.00
0.34
1,000
1,000
Mother’s Log income
Log-size of mother’s house
Daughter’s Log-income
Mother’s Age
Constant
Simulated data set, central scenario. Method: OLS
61
Figure 22: Density of relative variations of profits between RoT and optimal contracts
0
.05
.1
.15
Kernel density estimate
0
20
40
60
CwP vs RoT
kernel = epanechnikov, bandwidth = 1.0519
Source: Simulated data set, central scenario. RoT refers to Rule of
thumb contract. CwP (‘contract with profit’) refers to the optimal
contract with maximal profit π ∗ .
The next step is to look at the distribution of (πi∗ − πi (ρ))/πi (ρ), the relative variations of
profit. This distribution is presented in Fig. 22. The figure shows that profit can be multiplied
by more than 3 or 4 (or even more) as compared to the RoT contract. The relative profit increase
may be particularly large in some relatively less likely cases in which the mother is rich and the
daughter has a small income.
We now study the impact of some key parameters and compare profits in different scenarios.
First we look at the effect of increasing the mother’s degree of altruism. Fig. 23 gives the CDF of
ln π ∗ in the sample for the three values of γ: 1, 1.5 and 2. We see that the distributions obtained
with a higher value of the altruism parameter γ (almost) dominate the others in the sense of firstorder stochastic dominance (hereafter FOSD). Recall that distribution F dominates distribution
G in the sense of first-order stochastic dominance if the cdf of F is everywhere smaller or equal
than the cdf of distribution G. This confirms that a more altruist family is more profitable for
the bank.
If we now decrease the discount factor β0 , do we observe that the new distribution of π ∗
dominates the former in the sense of FOSD? The answer is yes: a less patient family (a lower
β0 ) is more profitable for the bank, as shown by Figure 24. Recall that β0 = β1 here. The FOSD
ranking is particularly obvious in the case of Fig. 24.
Finally, we computed the model’s solutions for several values of the CRRA risk-aversion
parameter σ. The values are .5,.6,.7,.8 and .9. Figure 25 plots the numerical results. There is
again a clear ranking of distributions. To a certain extent, a more risk-averse family is more
profitable, since the distribution for σ = .9 dominates the others. But the effect of risk aversion
62
0
.2
.4
.6
.8
1
Figure 23: Impact of altruism parameter γ on the cumulative distribution of profits
8
10
12
Gamma = 1
Gamma = 2
14
16
Gamma = 1.5
Source: Simulated data set, γ takes the three values: 1, 1.5 and
2. The figure represents the cumulative distribution function of
log-profits, ln(π ∗ ) for the three values of γ. First-order stochastic
dominance of distribution F over G is characterized by the property
that the cdf of F is everywhere smaller than the cdf of distribution
G.
0
.2
.4
.6
.8
1
Figure 24: Impact of time preference parameter β0 on the cumulative distribution of profits
8
10
12
Beta0 = 0.95
Beta0 = 0.97
14
Beta0 = 0.96
Beta0 = 0.98
Source: Simulated data set, β0 takes the four values: .95, .96, .97
and .98. The figure represents the cumulative distribution function
of log-profits, ln(π ∗ ) for the four values of β0 .
63
16
0
.2
.4
.6
.8
1
Figure 25: Impact of risk-aversion parameter σ on the cumulative distribution of profits
8
10
12
Sigma = 0.5
Sigma = 0.7
Sigma = 0.9
14
16
Sigma = 0.6
Sigma = 0.8
Source: Simulated data set, σ takes the 5 values: .5,.6,.7,.8 and
.9. The figure represents the cumulative distribution function of
log-profits, ln(π ∗ ) for the 5 values of σ.
is highly nonlinear here.
Our next step is to study the distribution of the daughter’s house size H1 , under the rule of
thumb contract (RoT), the Optimal Contract with zero profit (Cw0P), and the optimal contract
with maximal profit (CwP). Table 21 gives the summary statistics of the simulated distribution
of H1 under the three contracts. The optimal contract, with zero or maximal profit, permits
a substantial increase in the daughter’s house value. Figure 26 permits one to compare the
mother’s and the daughter’s house sizes. It is easy to see that under the optimal contract (with
or without profit), the distributions of H0 and H1 are much closer than under the RoT contract.
It is also interesting to see the distribution of relative variations in the daughter’s house size
between different contractual arrangements. The distribution of percentage variations ∆H1 /H1
are given in Figure 27. Let Ĥ1 (ρ) denote the daughter’s house area under the RoT contract, let
H1∗ denote the daughter’s house area under the optimal contract with maximal profit (CwP); let
finally H10 denote the daughter’s house size under the optimal contract with zero profit. The top
plot in Figure 27 gives the three smoothed densities of (H10 − H1∗ )/H1∗ , (H1∗ − Ĥ1 (ρ))/Ĥ1 (ρ) and
(H10 − Ĥ1 (ρ))/Ĥ1 (ρ). The first of these distributions is very concentrated as compared to the
other two, looking like a spike on the top plot. This shows that the optimal contract with profit
and with zero profits are in fact very close in terms of the daughter’s house size. The other two
distributions are the relative variations of house size under the optimal contract, with respect to
the RoT contract. The latter distributions are much more dispersed.
The middle picture in Fig. 27 shows that the bulk of the percentage variation between optimal
and RoT contracts is between 0% and 200%. There are huge variations, mainly due to the fact
64
Table 21: Size of the daughter’s house
Mean
Median
Standard deviation
Size of daughter’s house (RoT)
76
58
69
Size of daughter’s house (CwoP)
107
91
68
Size of daughter’s house (CwP)
97
82
63
Simulated data set, central scenario. House size in
square meters.
that the mother’s wealth is now transmitted to the daughter sooner than when it takes the form
of inheritance, and is also taken into account to assess the mortgage repayment ability of the
daughter. The bottom plot on Fig. 27 shows that most variations in real-estate investment, due
to the fact that profit would vary from its maximal value to zero, are in the range of 0 to 25%.
As a check, we run the regression of the log-house size on a number of covariates measuring
income and wealth. The results are given by Table 22. In these regressions based on artificial
data, we see that the mother’s income and real estate assets are significant only under the optimal
contracts (second and third columns of the table). It is also possible to explore the alternative
scenarios to study the effect of key parameters on the daughter’s real estate investment. First,
we look at the effect of increasing the mother’s degree of altruism. Fig. 28 gives the CDF of
H1∗ in the sample for the three values of γ: 1, 1.5 and 2. We see that the distributions obtained
with a higher value of the altruism parameter γ dominate the others in the sense of first-order
stochastic dominance. This confirms that a more altruist mother causes a larger real estate
investment on the part of the daughter. If we now increase the discount factor β0 , do we observe
that the new distribution of H1∗ dominates the former in the sense of FOSD? The answer is yes:
a more patient family (a larger β0 ) enables a larger home for the daughter, as shown by Figure
29, but the impact is small.
Finally, we computed the optimal house size for several values of the CRRA risk-aversion
parameter σ. The values are .5,.6,.7,.8 and .9. Figure 30 plots the numerical results, showing
cumulative distributions of H1∗ . There is again a clear ranking of these distributions. We conclude
that a more risk-averse daughter chooses a larger investment in real estate, in view of the clear
FOSD ranking.
We conclude this discussion of our illustrative simulations with a look at the distribution of
the relative variation of welfare W = U + γV in the central scenario. By definition, the variation
of welfare between the RoT contract and the optimal contract with maximal profit (CwP) is
zero (up to precision problems in the computation algorithm). Hence, we plot the distribution
of relative variations of W , in the central scenario, between the RoT contract and the optimal
zero-profit contract (Cw0P). This distribution is depicted in Figure 31. It provides indications
on the upper bound for welfare improvement attached to the optimal contract. The bulk of these
improvements are in the range of 5 to 10%, which is not negligible.
65
0
.002
.004
.006
.008
.01
Figure 26: Comparison of the mother’s and the daughter’s house-size distributions
0
100
200
300
400
500
x
Daughter's house (RoT)
0
.002
.004
.006
.008
.01
Mother's house
0
200
400
600
x
Daughter's house (CwP)
0
.002
.004
.006
.008
.01
Mother's house
0
200
400
600
x
Mother's house
Daughter's house (CwoP)
Source: Simulated data set, central scenario. House size in square
meters.
66
0
2
4
6
8
Figure 27: Distribution of percentage variation in daughter’s house size
-2
0
2
x
6
CwP vs RoT
0
.2
.4
.6
.8
CwoP vs RoT
CwP vs CwoP
4
-2
0
2
x
6
CwP vs RoT
0
2
4
6
8
CwoP vs RoT
4
0
.1
.2
.3
CwP vs CwoP
.4
.5
Source: Simulated data set, central scenario. House-size variations
in percentage. RoT refers to Rule of thumb contract. Cw0P means
optimal contract with zero profit. CwP means optimal contract
with maximal profit π ∗ .
67
Table 22: Regression of the daughter’s log-house-size on log-incomes and log-size of the mother’s
house
ln H1 under RoT
ln H1 under CwoP
ln H1 under CwP
-0.000
0.357***
0.374***
(0.000)
(0.004)
(0.001)
0.000
0.085***
0.042***
(0.001)
(0.008)
(0.002)
1.001***
0.440***
0.481***
(0.000)
(0.004)
(0.001)
-0.010***
0.003***
0.006***
(0.000)
(0.001)
(0.000)
-6.314***
-4.845***
-5.525***
(0.004)
(0.055)
(0.015)
R2
1.00
0.98
1.00
1,000
1,000
1,000
Mother’s log-income
Log-size of mother’s house
Daughter’s log-income
Age of the mother
Constant
Simulated data set, central scenario. Method: OLS
0
.2
.4
.6
.8
1
Figure 28: Impact of altruism parameter γ on the cumulative distribution of house area
0
200
400
Gamma = 1
Gamma = 2
600
800
Gamma = 1.5
Source: Simulated data set, γ takes the three values: 1, 1.5 and 2. The figure
represents the cumulative distribution function of the daughter’s house area in
square meters, H1∗ for the three values of γ. First-order stochastic dominance
of distribution F over G is characterized by the property that the cdf of F is
everywhere smaller than the cdf of distribution G.
68
Figure 29: Impact of time preference parameter β0 on the cumulative distribution of the daugh-
0
.2
.4
.6
.8
1
ter’s house size
0
200
400
Beta0 = 0.95
Beta0 = 0.97
600
Beta0 = 0.96
Beta0 = 0.98
Source: Simulated data set, β0 takes the four values: .95, .96, .97
and .98. The figure represents the cumulative distribution function
of the daughter’s house, H1∗ for the four values of β0 .
Figure 30: Impact of risk-aversion parameter σ on the cumulative distribution of the size of the
0
.2
.4
.6
.8
1
daughter’s house
0
200
400
600
Sigma = 0.5
Sigma = 0.7
Sigma = 0.9
800
Sigma = 0.6
Sigma = 0.8
Source: Simulated data set, σ takes the 5 values: .5,.6,.7,.8 and
.9. The figure represents the cumulative distribution function of
log-house-size, ln(H1∗ ) for the 5 values of σ.
69
1000
Figure 31: Distribution of percentage variation in family welfare between rule of thumb and
0
10
20
30
optimal zero-profit contract
0
.05
.1
Percentage variation of family welfare (CwoP vs RoT)
.15
Source: Simulated data set, central scenario. The figure represents the smoothed
density function of percentage variations in family welfare, W = U + V , between
Cw0P and RoT contract.
7
Concluding Remarks. Design of Family Contracts, Behavioral
Obstacles and Marketing
The above theory and the associated numerical analysis, based on a fully specified and calibrated
version of our model, both suggest that there is room for improvements in the efficiency of family
arrangements, relative to standard practices of inter-vivos gifts and inheritance. Our contribution
is not to emphasize the importance of equity release instruments for the wealth management of the
elderly. More precisely, our contribution is to study the combination of equity release contracts
with more general insurance and household investment strategies, involving several members of
a family and different generations. A banker and an insurer can jointly improve the efficiency of
family arrangements, in particular, because they can radically improve risk-sharing within the
family. The banker-insurer can immunize the family against real-estate price risk and at the
same time immunize the heirs (children or spouse, or both) against longevity risk, by means of
structured inter-temporal transfers between parents and children. The banker-insurer acts as a
middleman for family members. The gifts to children being in fact carried out by the financial
intermediary, they can be made independent of the event of a parent’s death. Under the terms
of a family contract, the banker-insurer supplies the family with a bundle of services: liquidity,
various kinds of loans and insurance; the financial intermediary satisfies the demand for bequest
and gifts, due to the altruist motives of the parents, on behalf of the parents. The intermediated
family arrangement can be interpreted as a bundle of bilateral contracts with the bank.
70
Another important question is the possibility of removing the behavioral obstacles to the use
of equity-release instruments and annuitization of wealth. If the elderly have a bequest motive
(based on pure altruism or some taste for giving), they could annuitize their wealth partially,
and ring-fence part of their assets for inheritance. But if the children are just slightly impatient,
then the money should be transmitted as soon as possible in the form of an inter-vivos gift. This
could happen, say, at the latest of the two moments when the parents have completely repayed
the mortgage on their residence and when their children reach adulthood and want to settle. In
practice, given the facts studied in Section 2 above, the elderly members of the family may in fact
wish to help their grandchildren, because they are typically still young when their own children
want to buy a house. Home reversion plans or intermediated viager contracts can do the job of
transmitting the present value of the parent’s real estate assets sooner than the death of the last
surviving member of the parent’s household. It’s rational to annuitize the real-estate assets by
means of a viager or home reversion contract only if the family seniors are at the same time well
insured against the need for long-term care, i.e., the risk of moving to a nursing home or the risk
of a sharp increase in health expenditures. Our model should and could be extended to take into
account the long-term care and health risks, at the cost of additional complexity. But it’s clear
that the banker-insurer can price the relevant risks with a reasonable degree of accuracy and
solve the equity-release and annuitization problems completely. What remains is to structure
the timing and sharing of the estate among family members, and in so doing, to insure the heirs
against the father’s (or the mother’s) longevity risk. The banker-insurer can do something that
the mother or the father could not do without his help by just selling his (her) house on a viager
or home reversion market. On her own, the mother could share the annuity value of the viager
sale of her house among her children. This move would insure the whole family against future
house-price risk. But the money distribution, i.e., transfers to children would stop when the
mother passes away. This is a risk if the children still have to repay mortgages at this moment.
The banker-insurer can commit to continue subsidizing the children’s asset building efforts during
a time span that is completely independent of the mother’s longevity. This is the key element in
our theory of intermediated family contracts. There are of course some tax optimization aspects
in the design of these contracts. Since the intermediated family arrangement can be interpreted
as a bundle of bilateral contracts with the bank, it is unclear whether the transfers of purchasing
power between family members implemented by the bundle of contractual arrangements can be
interpreted (in legal terms) as gifts, or if they must be viewed as cross-subsidies between several
categories of the bank’s clients, or even simply as promotional pricing aimed at new customers.
It may be that inter-vivos gifts are taxed, in a given country, while cross-subsidies between clients
or products are in fact a legally admissible form of (third-degree) price discrimination.
Assuming that the long-term care insurance problem has been solved, it is reasonable to
expect that an intermediated family contract would help removing the behavioral obstacles to
annuitization of the elderly’s wealth, because annuitization becomes part of an encompassing
solution to a family problem. Of course, further empirical research would be needed to really
estimate the extent to which the latter claim is true.
71
8
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9
Appendix: Proofs
Proof of Proposition 1. Let λ be the Lagrange multiplier of the actuarial resource constraint
(4) above; let νt be the multiplier of Zt ≥ 0; let µ be the multiplier of constraint
H0 − z1 − z2 − z0 ≥ 0;
Taking derivatives with respect to z0 first, assuming interiority, i.e., z0∗ > 0, we find the necessary
conditions
λXτb =µ;
(43)
µ(H0 − z0 − z1 − z2 ) =0;
(44)
λ ≥ 0;
µ ≥ 0;
(45)
It follows immediately that if λ > 0 and τb > 0 then µ > 0 and H0 = z0 + z1 + z2 . But these
conditions do not pin down z0∗ .
To do this, we write the first order conditions with respect to z1 and z2 . This yields,
0
0
−v0h
(H0 − z1 , H0 ) − β0 pv0h
(H0 − z1 − z2 , H0 ) = λ[(1 − τb )X − q1 ] − ν1 + µ;
q2
0
(H0 − z1 − z2 , H0 ) − λ(1 − τb )X = − λ
−β0 pv0h
(1 − τb (1 − p)) − ν2 + µ;
(1 + r)
(46)
(47)
ν1 z1 = ν2 z2 = 0;
(48)
µ(H0 − z0 − z1 − z2 ) = 0;
(49)
(µ, ν1 , ν2 ) ≥ 0.
(50)
But using λXτb = µ we find,
0
(H0 , H0 ) + λ(X − q1 ) = ν1 > 0;
(1 + β0 p)v0h
q2
0
β0 pv0h (H0 , H0 ) + λ X −
(1 − τb (1 − p)) = ν2 > 0.
(1 + r)
(51)
(52)
These inequalities will always be true if the left derivative of v0 at point (H0 , H0 ) is large enough.
It follows, that z1∗ = z2∗ = 0. It is easy to check (see below) that λ∗ > 0. As a consequence, since
τb > 0 by assumption, we have H0 = z0∗ . Q.E.D.
Proof of Proposition 2. We first take derivatives with respect to H1 , assuming interiority,
that is, H1∗ > 0. We obtain the following first-order condition.
v10 (H1 + h1 ) + pβ1 v10 (H1 + h2 ) + pβ12 v10 (H1 + h3 )
+(1 − p) β1 v10 (H1 + ĥ2 ) + β12 v10 (H1 + ĥ3 )
λ
q4
=
q1 −
.
γ
(1 + r)3
76
(53)
If we take derivatives with respect to ht and ĥt , we get the following conditions
γv10 (H1 + h1 ) ≤ λR1 ;
h1 ≥ 0
R2
; h2 ≥ 0
(1 + r)
R3
γβ12 v10 (H1 + h3 ) ≤ λ
; h3 ≥ 0
(1 + r)2
R2
γβ1 v10 (H1 + ĥ2 ) ≤ λ
; ĥ2 ≥ 0
(1 + r)
R3
γβ1 v10 (H1 + ĥ3 ) ≤ λ
; ĥ3 ≥ 0.
(1 + r)2
γβ1 v10 (H1 + h2 ) ≤ λ
(54)
(55)
(56)
(57)
(58)
where the inequalities become equalities if the corresponding h∗t > 0 or ĥ∗t > 0. Substituting
the above inequalities in the first-order condition for an optimal H1 above, we find the arbitrage
inequality given in the statement of Proposition 2 above. When h∗t > 0 and ĥ∗t > 0, the above
inequalities become equalities and the arbitrage equation holds exactly. If h∗t = 0 and ĥ∗t = 0,
the first-order condition for H1∗ boils down to the expression given as point 3 in the statement
of Proposition 2. Q.E.D.
Proof of Proposition 3. The conditions stated in Proposition 3 above are straightforward
consequences of the first-order optimality conditions, applied to consumption. Q.E.D.
Proof of Proposition 4. To prove this proposition, let αt be the Lagrange multiplier of the
sign constraint xt ≥ 0. The Kuhn-Tucker conditions for an optimal gift x∗t can be written as
follows. We have αt ≥ 0, and αt xt = 0, together with the conditions,
α1 = λτg ;
α2 =
λpτg
.
(1 + r)
(59)
(60)
It follows from this, that, if τg > 0, then it must be that αt∗ > 0, implying x∗t = 0. Q.E.D.
Proof of Proposition 6. To prove this proposition we write the Kuhn and Tucker necessary
conditions for optimality (maximizing the mother’s expected utility subject the ARC 38. We introduce Lagrange multipliers, denoted νi , i = 0, . . . , T for the zi variables and the complementary
slackness conditions zi ≥ 0, νi zi = 0. We introduce a Lagrange multiplier µ for the constraint
ΣT0 zt ≤ H0 and we have the condition µ(H0 −ΣT0 zt ) = 0. Let finally λ be the Lagrange multiplier
of the actuarial resource constraint 38. The first-order condition for z0 , assuming z0 > 0, is the
following,
λX = λ
T
X
β T (1 − τb )pT qT +1 + µ.
T =1
Using the expression for X, this can be rewritten,
λτb
T
X
β T pT qT +1 = µ.
T =1
77
(61)
From this equation, we derive that λ > 0 implies µ > 0, and therefore, H0 = ΣTt=0 zt (the mother’s
house is entirely sold before death).
We now write the first-order conditions for the optimality of (z1 , . . . , zT ). For t > 0 we find,
!
P
T
τ
X
X
∂ τθ=1 zθ
τ −1
0
β0 Sτ v0 H0 −
zθ
−
∂zt
τ =1
θ=1
!
PT
T
X
z
∂
θ=0 θ
+νt + λβ t−1 St qt − λ
= µ.
pT β T qT +1 (1 − τb )
∂zt
T =1
Now, given that
∂

1 if
z
θ=1 θ
=
0 if
∂zt
Pτ
t≤τ
t > τ,
the first-order condition for (zt ), t > 0 can be rewritten
!
T
τ
T
X
X
X
µ=−
β0τ −1 Sτ v00 H0 −
zθ + νt + λβ t−1 St qt − λ(1 − τb )
pT β T qT +1 .
τ =t
T =t
θ=1
This can again be rewritten,
νt =
T
X
β0τ −1 Sτ v00
H0 −
τ =t
τ
X
!
zθ
+ µ + λ(1 − τb )
T
X
pT β T qT +1 − λβ t−1 St qt .
(62)
T =t
θ=1
Given 61, we know that µ > 0. Thus, all the terms on the right hand side of 62 are positive
except the last one. If the left-derivative of v0 at point H0 is large enough, then, we have νt > 0
for all t > 0 at the optimum. By complementary slackness conditions, we conclude that zt = 0
at the optimum for all t > 0. This in turn implies (with µ > 0) that z0 = H0 at the optimum:
the mother sells her entire house in the form of a viager contract at time t = 1.
Next, assuming interiority, that is H1 > 0 at the optimum, we take the derivative of the
Lagrangian with respect to H1 and we find the condition,
TX
+1
t=1
β1t−1 St v10 (H1 + ht ) +
T
X
T =1
TX
+1
pT
λ
(q1 − qT +2 β T +1 ).
γ
β1t−1 v10 (H1 + ht (T )) =
t=T +1
(63)
Let now ηt (resp. ηt (T )) denote the Lagrange multiplier of the sign constraint ht ≥ 0 (resp.
ht (T ) ≥ 0). Kuhn and Tucker’s Theorem tells us that these multipliers are nonnegative. Taking
partial derivatives of the Lagrangian with respect to ht and ht (T ) and equating them to zero
yields the following necessary conditions,
γβ1t−1 St v10 (H1 + ht ) + ηt = λβ t−1 St Rt ,
(64)
for all t = 1, . . . , T , and with the condition ηt ht = 0, and,
γβ1t−1 v10 (H1 + ht (T )) + ηt (T ) = λβ t−1 pT Rt ,
(65)
for t > T and all T = 1, . . . , T with the condition ηt (T )ht (T ) = 0. Now, we substitute 64 and
65 in 63 and we eliminate the terms involving v10 . This yields,
q1 ≤ qT +2 β T +1 +
TX
+1
β1t−1 St Rt +
t=1
TX
+1
t=2
78
β1t−1 Rt
t−1
X
T =1
pT .
using the fact that S1 = 1 − P1 = 1 and ST +1 = 0, we finally obtain the arbitrage inequality,
q1 ≤
TX
+1
β t−1 Rt + qT +2 β T +1 .
(66)
t=1
Inequality 66 becomes an equality if the daughter rents at least part of her housing area, since
ht > 0, ht (T ) > 0 implies ηt = 0, ηt (T ) = 0.
If 66 holds, then ht = 0, ht (T ) = 0 is the solution, and the daughter’s house size H1 is
determined by the following equation, derived from 63. Assuming that the effective price of
property q1 − qT +2 β T +1 is positive, we have,
v10 (H1 ) =
λ (1 − β1 )
q1 − qT +2 β T +1 .
T
+1
γ (1 − β1 )
(67)
Taking the partial derivatives of the Lagrangian with respect to consumption variables and
equating to zero, we find u0 (c01 ) = λ > 0 since S1 = 1 and
β0t−1 u0 (c0t ) = λβ t−1
(68)
γβ1t−1 u0 (c1t ) = λβ t−1
(69)
γβ1t−1 u0 (c1t (T )) = λβ t−1 .
(70)
From the latter equations, we easily derive points 1-4 in the statement of Proposition 6 and the
full insurance property.
Finally, let αt be the Lagrange multiplier of the sign constraint on gifts xt , that is, xt ≥ 0,
αt ≥ 0 and αt xt = 0. Kuhn-Tucker conditions yield αt = λβ t−1 St τg . But λτg > 0 implies αt > 0
and therefore xt = 0 at the optimum. Q.E.D.
79

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